Adomian - Solving_Frontier_Problems_of_Physics

Solving Frontier Problems ofPhysics: TheDecomposition Method by George Adomian ‘Kiuwer Academic Publishers Fundamental Theories ofPhysics Solving Frontier Problems ofPhysics: The Decomposition Method H || | i { Fundamental Theories ofPhysics AnInternational Book Series onTheFundamental Theories ofPhysics:TheirClarification, Development andApplication Editor: ALWYN VAN DER MERWE University ofDenver,US.A. Editorial Advisory Board: ASIM BARUT, University ofColorado, US.A. BRIAN D.JOSEPHSON, University ofCambridge, U.K. CLIVEKILMISTER. University ofLondon,UK. GUNTER LUDWIG. Philipps-Universitat, Marburg, Germany NATHAN ROSEN.Israe!InstituteofTechnology. IsraelMENDEL SACHS.StareUniversity ofNewYorkaiBuffalo,US.A. ABDUS SALAM. International Centre forTheoretical Physics, Trieste, Italy HANS-JURGEN TREDER, Zentralinstitut fiurAstrophysik derAkademie der Wissenschaften. Germany Volume 60 Solving Frontier Problems ofPhysics: The Decomposition Metho Gengdoin na we KLUWER ACADEMIC PUBLISHERS LitaryofCongressCataloging PubicationData sei ethetdStbienn can. punees Be ts wee Sai eeacae sasseees ISBN 0.7923-2640x Published byKluwer Academic Pabishes PO Bon 17,300 AA Dordrecht, The Netherlands, ower Academie Publishers incorporates thepublishing propemnes of D.Reide, Marinus Nit, DeW,nk andMTP Press. Sold and distributed inheU.S.A. and Canada byRiuwer Academie Publishers TO!Pip Dave, Norwell Ma62061. US.A Inallothercounties, sold and istbuted trKiower Academic Publishers Grup. PO Bon 322 3500 AH Device The Netherlands, a”oa Oe C ‘ALigh Reserved61904KlowerAcademicPublishesNoparothe material peered Bythiscopyright notice may bereproduced or Uilzd inanyform orbyaymean lectons omeshanie Inclugingphotoeopving ecording obyanyinformation sorage an raieval atm, wihos writen formisaoy fam iheopsngh ot, Pred inine Netherlands INMEMORY OFMY FATHER AND MOTHER HAIG AND VARTUBI ADOMIAN Soe) gine oll Lt NG46142 ofl ae EARLIER WORKS BYTHE AUTHOR Applied Stochastic Processes, Academic Press, 1980. Stochastic Systems, Academic Press, 1983; also Russian transl. ed.HG.Volkova, MirPublications, Moscow, 1987.Partial Differential Equations with R.E.Bellman, D.Reidel Publishing Co 1985, Nonlinear Stochastic Operator Equations, Academic Press, 1986 Nonlinear Stochastic Systems Theory andApplications toPhysics, Kluwer ‘Academic Publishers, 1989. TABLE OF CONTENTS PREFACE ix FOREWORD xi CHAPTER 1 ON MODELLING PHYSICAL PHENOMENA 1 “CHAPTER 2 THE DECOMPOSITION METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS 6 CHAPTER 3 THE DECOMPOSITION METHOD INSEVERAL DIMENSIONS 22 CHAPTER 4 DOUBLE DECOMPOSITION 69 CHAPTER §—MODIFIED DECOMPOSITION 11s CHAPTER 6 APPLICATIONS OF MODIFIED DECOMPOSITION 154 CHAPTER 7 DECOMPOSITION SOLUTIONS FOR NEUMANN BOUNDARY CONDITIONS 190 CHAPTER 8 INTEGRAL BOUNDARY CONDITIONS 196 “CHAPTER 9 BOUNDARY CONDITIONS ATINFINITY 2u1 CHAPTER 10 INTEGRAL EQUATIONS 224 CHAPTER 11 NONLINEAR OSCILLATIONS INPHYSICAL SYSTEMS 228 CHAPTER 12 SOLUTION OF THE DUFFING EQUATION 236 CHAPTER 13. BOUNDARY-VALUE PROBLEMS WITH CLOSED IRREGULAR CONTOURS OR SURFACES 288 CHAPTER 14 APPLICATIONS IN PHYSICS 302 APPENDIX I PADE AND SHANKS TRANSFORMS 338 APPENDIX I] ON STAGGERED SUMMATION OF DOUBLE DECOMPOSITION SERIES 348 APPENDIX III CAUCHY PRODUCTS OF INFINITE SERIES 350 INDEX 352 - vii PREFACE Idiscovered thevery interesting Adomian method andmetGeorge Adomian himself some years agoataconference held intheUnited States. This new technique was very surprising forme,anapplied mathematician, because it allowed solution ofexactly nonlinear functional equations ofvarious kinds (algebraic, differential, partial differential, integral...) without discretizing the equations orapproximating theoperators. Thesolution when itexists isfound inarapidly converging series form, andtime andspace arenotdiscretized. At thistime animportant question arose: why does this technique, involving special kinds ofpolynomials (Adomian polynomials) converge? Iworked on thissubject with some young colleagues atmyresearch institute andfound that, itwaspossible toconnect themethod tomore well-known formulations where classical theorems (fixed point theorem, substituted series, ..)could beused. Ageneral framework fordecomposition methods haseven been proposed by Lione! Gabet, oneofmyresearchers who hasobtained aPh.D. thesis onthis subject. During thisperiod afruitful cooperation hasbeen developed between George Adomian andmyresearch institute. Wehave frequently discussed advances anddifficulties andweexchange ideas andresults. With regard tothisnew book, Iamvery impressed bythequality andthe importance ofthework, inwhich theauthor uses thedecomposition method forsolving frontier problems ofphysics. Many'concrete problems involving differential and partial differential equations (including Navier-Stokes equations) aresolved bymeans ofthedecomposition technique developed by Dr.Adomian. The basic ideas areclearly detailed with specific physical examples sothatthemethod canbeeasily understood andused byresearchers ofvarious disciplines. One ofthemain objectives ofthismethod istoprovide asimple andunified technique forsolving nonlinear functional equations. Ofcourse some problems remain open, Forinstance, practical convergence may beensured even ifthehypotheses ofknown methods arenotsatisfied. ‘That means thatthere stillexist opportunities forfurther theoretical studies to bedone bypure orapplied mathematicians, such asproving convergence in More general situations. Furthermore, itisnotalways easy totake into account theboundary conditions forcomplex domains. Inconclusion, Ithink that this book isafundamental contribution tothe theory and practice ofdecomposition methods infunctional analysis. It rc completes andclarifies theprevious book oftheauthor published byKluwer in 1989. The decomposition method hasnow lostitsmystery butithaswon in seriousness and power. Dr. Adomian istobecongratulated forhis fundamental contribution tofunctional and numerical analysis ofcomplex systems, Yves Cherruault Professor Director ofMedimat Université Pierre etMarie Curie (Paris VD) Paris, France September 9,1993 FOREWORD ‘This book isintended forresearchers and(primarily graduate) students of physics, applied mathematics, engineering, and other areas such as biomathematics and astrophysics where mathematical models ofdynamical systems require quantitative solutions. Amajor partofthebook deals with the necessary theory ofthedecomposition method and itsgeneralizations since earlier works. Anumber oftopics arenotincluded here because they were dealt with previously. Some ofthese aredelay equations, integro-differential equations, algebraic equations and large matrices, comparisons of decomposition with perturbation and hierarchy methods requiring closure approximation, stochastic differential equations, andstochastic processes [1]. Other topics hadtobeexcluded duetotime andspace limitations aswell asthe objective ofemphasizing utility insolving physical problems. Recent works, especially byProfessor Yves Cherruault injournal articles andbyLionel Gabet inadissertation, have provided arigorous theoretical foundation supporting the general effectiveness ofthe method of decomposition. Theauthor believes thatthismethod isrelevant tothefield of mathematics aswell asphysics because mathematics hasbeen essentially a linear operator theory while wedeal with anonlinear world. Applications have shown that accurate andeasily computed quantitative solutions can be determined fornonlinear dynamical systems without assumptions of“small” nonlinearity orcomputer-intensive methods. The evolution oftheresearch hassuggested atheory tounify linear and nonlinear, ordinary orpartial differential equations forsolving initial or boundary-value problems efficiently. Assuch, itappears tobevaluable inthe background ofapplied mathematicians and theoretical ormathematical physicists. Animportant objective forphysics isamethodology forsolution of dynamical systems—which yields verifiable andprecise quantitative solutions tophysical problems modelled bynonlinear partial differential equations in space andtime, Analytical methods which donotrequire achange ofthemodel equation into mathematically more tractable, butnecessarily less realistic tepresentation, areofprimary concern. Improvement ofanalytical methods would inturn allow more sophisticated modelling and possible further Progress. Thefinaljustification oftheoriesofphysics isinthecorrespondence ofpredictions with nature rather than inrigorous proofs which may well ai Foreworo restrict thestated problem toamore limited universe. Thebroad applicability ofthemethodology isadividend which may allow anew approach to ‘mathematics courses asweil asbeing useful forthephysicists who will shape ourfuture understanding oftheworld. Recent applications byagrowing community ofusers have included areas such asbiology andmedicine, hydrology, andsemiconductors. Intheauthor's opinion thismethod offers afertile field forpure mathematicians andespecially fordoctoral students looking fordissertation topics. Many possibilities are included directly orindirectly. Some repetition ofobjectives andmotivations (forresearch ondecomposition andconnections with standard methods) was believed tobeappropriate tomake various chapters relatively independent and permit convenient design ofcourses fordifferent specialties andlevels, Partial differential equations arenow solved more efficiently, with less computation, than intheauthor's earlier works. The Duffing oscillator and other generic oscillators aredealt with indepth. The lastchapter concentrates onanumber offrontier problems. Among these arethe Navier-Stokes equations, theN-body problem, and theYukawa-coupled Klein-Gordon- Schrodinger equation. The solutions ofthese involve nolinearization, perturbation, orlimit onstochasticity. TheNavier-Stokes solution [2]differs from earlier analyses [3].The system isfully dynamic, considering pressure changing asthevelocity changes. Itnow allows high velocity and possible prediction oftheonset ofturbulence. The references listed arenotintended tobeanexhaustive oreven apartial bibliography ofthevaluable work ofmany researchers inthese general areas. Only those papers arelisted which were considered relevant totheprecise area and method treated. (New work isappearing now atanaccelerating rate by many authors forsubmission tojournals orfordissertations and books. A continuing bibliography could bevaluable tofuture contributors andreprints received bytheauthor willberecorded forthispurpose.) The author appreciates theadvice, questions, comments, andcollaboration ofearly workers inthis field such asProfessors RE. Bellman, N.Bellomo, Dr.R.MCarty, and other researchers over theyears, theimportant work by Professor Yves Cherruault onconvergence andhismuch appreciated review of theentire manuscript, thesupport ofmyfamily, andtheediting andvaluable contributions ofcollaborator andfriend, Randolph Rach, whose insights and willingness toshare histime andknowledge ondifficult problems have been animportant resource. The book contains work originally typeset byArlette Foreworo sisi Revells andKarin Haag. Thecamera-ready manuscript waspreparedwiththe dedicated effort ofKarin Haag, assisted byWilliam David. Laura andWilliam David assumed responsibility foroffice management sothatresearch results could beaccelerated. Computer results ontheDuffing equation were obtained byDr.McLowery Elrod with thecooperation oftheNational Science Center Foundation headed byDr.Fred C.Davison, who haslong supported this work. Gratitude isdue toRonald E.Meyers, U.S. Army Research Laboratories, White Sands Missile Range, who supported much ofthis research andalso contributed tosome ofthedevelopment. Thanks arealso due totheOffice ofNaval Research, Naval Research Laboratories, and Paul Palo oftheNaval Civil Engineering Laboratories, who have supported work directed toward applications aswell asintensive courses atNRL andNCEL. ‘The author would also liketothank Professor Alwyn Van derMerwe ofthe University ofDenver forhisencouragement thatledtothisbook. Most ofall, theunfailing support bymywife, Corinne, aswell ashermeticulous final editing, isdeeply appreciated, G.Adomian REFERENCES 1.G.Adomian,StochasticProcesses, Encyclopedia ofSciencesandTechnology. 16,2nd ed., Academic Press (1992). 2.G, Adomian, An Analytic Solution tothe Stochastic Navier-Stokes System.Foundations ofPhysics.2,(831-834)(July1991). 3.G.Adomian, Nonlinear Stochastic SystemsTheoryandApplications toPhysics,Kluwer (192-216) (1989). CHAPTER 1 ON MODELLING PHYSICAL PHENOMENA Ouruseoftheterm“mathematical model”or“model”willrefertoasetof consistent equations imended todescribe theparticular features orbehavior of aphysical system which weseck tounderstand. Thus, wecanhave different modelsofthesystemdependent onthequestions ofinterest and onthe features relevant tothose questions. Toderive anadequate mathematical description with aconsistent setofequations andrelevant conditions, weclearly must have inmind apurpose orobjective andlimit theproblem toexclude factors irrelevant toour specific interest. Webegin byconsidering thepertinent physical principles whichgovernthephenomena ofinterest alongwiththe constitutive properties ofmaterial with which thephenomena may interact. Depending ontheproblem, amodel may consist ofalgebraic equations, imegral equations, orordinary, partial, orcoupled systems ofdifferential equations. Theequations canbenonlinear andstochastic ingeneral with linear ordeterministic equations being special cases. (Insome cases, wemay have delays aswell.) Combinations ofthese equations such asintegro-differential equations also occur. Amodel using differential equations must also include theinitial/boundary conditions. Since nonlinear andnonlinear stochastic equations areextremely sensitivetosmallchangesininputs,parameters, orinitialconditions, solutions may change rather radically with such changes. Consequently, exact specification ofthemodel issometimes notasimple matter. Prediction of future behavior istherefore limited bytheprecision oftheinitial state. When significant nonlinearity ispresent, small changes (perhaps only 1%) inthe system may make possible oneormany different solutions. Ifsmall but appreciable randomness, or,possibly, accumulated round-off error initerative calculation ispresent, wemay observe arandom change from onesolution to another—an apparently chaotic behavior. Tomodelthephenomena, process,orsystemofinterest, wefirst isolate the relevant parameters. From experiments, observations, and known relationships, weseek mathematical descriptions intheform ofequations which wecan then solve fordesired quantities. This process isneither universal norcanittake everything intoaccount; wemust tailor themodel tofit 1 2 Chapren } thequestions towhich weneed answers andneglect extraneous factors. Thus amodel necessarily excludes theuniverse external totheproblem andregion of imerest tosimplify asmuch aspossible, andreasonably retain only factors relevant tothedesired solution. Modelling isnecessarily acompromise between physical realism andour ability tosolve theresulting equations. Thus, development ofunderstanding based onverifiable theory involves both modelling andanalysis. Any incentive formore accurate orrealistic modelling islimited byourability tosolve the equations; customary modelling uses restrictive assumptions sothat well- known mathematics canbeused. Our objective istominimize oravoid altogether this compromise formathematical tractability which requires linearization andsuperposition, perturbation, etc., andinstead, tomodel the problem with itsinherent nonlinearities andrandom fluctuation oruncertain data. ‘Wedothisbecause thedecomposition micthod isintended tosolve nonlinear and/or stochastic ordinary orpartial differential equations, integro-differential equations, delay equations, matrix equations, etc., avoiding customary restrictive assumptions and methods, toallow solutions ofmore realistic models. Ifthedeductions resulting from solution ofthismodel differ from accurate observation ofphysical reality, then thiswould mean thatthemodel is apoor oneandwemust re-model theproblem. Hence, modelling andthe solution procedure ought tobeapplied interactively. Since wewill bedealing with alimited region ofspace-time which isofinterest totheproblem athand. wemust consider conditions ontheboundaries oftheregion tospecify the problem completely. Ifweareinterested indynamical problems such asa process evolving over time, then wemust consider solutions astime increases from some initial time: i.¢., wewill require initial conditions. We will be interested generally indifferential equations which express relations between functions and derivatives. These equations may involve use offunctions, ordinary orpartial derivatives, andnonlinearities andeven stochastic processes todescribe reality. Also, ofcourse. initial andboundary conditions must be specified tomake theproblem completely determinable. Ifthesolution istobevalid, itmust satisfy thedifferential equation andthe properly specified conditions, soappropriate smoothness must exist. Wehave generally assumed that nonlinearities areanalytic but will discuss some exceptions inalater chapter. Anadvantage, other than thefactthatproblems areconsidered more realistically than bycustomary constraints. isthat Ow MODELLING PHYSICAL PHENOWENA 2 solutions are not obtained here bydiscretized methods: solutions are continuous andcomputationally much more efficient asweshall see.Ifwe candeal with aphysical problem asitis, wecanexpect auseful solution, i... one inwhich themathematical results correspond toreality. Ifourmodel is poor because thedataarefound from measurements which have some error, it isusual torequire thatasmall change inthedata must lead toasmall change in thesolution. This does notapply tononlinear equations because small changes ininitial data can cause significam changes inthesolution, especially in stochastic equations. This isaproblem ofmodelling. Ifthedata arecorrect and. the equation properly describes theproblem, we expect acorrect and convergent solution. ‘The initial/boundary conditions foraspecific partial differential equation, needlesstosay,cannotbearbitrarily assigned: theymustbeconsistent withthe physical problem being modelled. ‘Suppose weconsider asolid body where u(x,y,z.t) represents atemperature atx,y,z attime t.Ifweconsider avolume Vwithin thebody which isbounded byasmooth closed surface $andconsider thechange ofheatinVduring an interval (t,t), wehave, following thederivation ofN.S. Koshlyakov, MM. Smimov, and E.B. Gliner [1] QafaJfelxy.2)22 as a an where nisthenormal to$inthedirection ofdecreasing temperatures andkis theinternalheatconductivity, apositivefunctionindependent ofthedirectionofthenormal,Theamountofheattochangethetemperature ofVis =f" affoe2evee IY Poe where c(x,y,2) isthespecific heat andp(x.y,2) isthedensity. Ifheat sources with density g(x,y.2.1) exist inthebody, wehave Q=fafffeuy.zuav SinceQ:=Q,+Qs,itfollowsthat ‘ Charcen | cpot=div(kgradu)+ at Ifcpand kareconstants, wecan write a’=k/ep and f(x,y,z,1) = g(x.yz.0/ep .Then OU_ ag?eavuse (which neglects heat exchange between Sandthesurrounding medium). Now todetermine asolution, werequire thetemperature ataninitial instant 1u(x,y,z,t =0)andcither thetemperatures atevery point ofthesurface orthe heat flow onthesurface. These areconstraints orcommonly, theboundary conditions. Ifwedonotneglect heat exchange tothesurrounding medium which isassumed tohave uniform temperature up,athird boundary condition canbewritten as@(u-u,)=-kdu/dn|, (ifweassume thecoefficient of exchange isuniform forallofS). ‘Thus thesolution must satisfy theequation, theinitial condition, andoneof the above boundary conditions orconstraints which make the problem specific. We have assumed aparticular model which isformulated using fundamental physical laws such asconservation ofenergy. sotheinitial distribution mustbephysically correctandnotarbitrary. Ifitiscorrect,itleads toaspecific physically correct solution, The conditions andtheequation must. beconsistent and physically correct. The conditions must besmooth, bounded, andphysically realizable. The initial conditions must beconsistent with theboundary conditions andthemodel.Thederived“solution” isverified tobeconsistent with themodel equation andtheconditions andistherefore the solution, NOTE: Koshlyakov, et.al,[1]state that wemust specify u(t=0)within thebody and oneoftheboundary conditions such asuonS.However Sis notinsulated from thebody. Theinitial condition u(t=0)fixes uonSalso if Surroundings areignored. Itseems that either one ortheother should be enough inaspecific problem and ifyou give both, they must beconsistent with each other and themodel (equation). The same situation arises when, €.g., inasquare orrectangular domain, weassign boundary conditions onthe four sides, which means thatphysically wehave discontinuity atthecomers (OW MODELLING PHYSICAL PHENOMENA s REFERENCE 1. NS. Koshlyakov, M.M.Smirnov, and E.B. Gliner, Differential Equations of Mathematical Physics, North Holland Publishers (1964) SUGGESTED READING 1.¥.Cherruault, Mathematical Modelling inBiomedicine, Reidel(1986).2. R.P. Feynman, R.B.Leighton, and M.Sands, The Feynman Lectures onPhysics. ‘Addison-Wesley (1965). 3. I.S.Sokolnikoff and RM. Redheffer, Mathematics ofPhysics and Modern Engineering, 2nded... McGraw-Hill (1966). CHAPTER 2 THE DECOMPOSITION METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS A on ‘AciitiCally important probleminfrontierscienceandtechnology istheur.physically gorrect solution ofnonlinear and/or stochastic systems modelled by;differGhtacoF |raditterentat equationsfor"gerféral_initial/boundary ©,” conditions.yb CEURGRR ~ ‘The usual procéfures ofanalysis necessarily change such problems in essential waysinorder tomake them mathematically tractable byestablished methods, Unfortunately thesechanges necessarily change thesolutions; therefore, they can deviate, sometimes seriously, from theactual physical behavior. These procedures include linearization techniques. perturbation methods, andrestrictions onthenatureandmagnitude ofstochastic processes. The avoidance ofthese limitations sothat physically correct solutions canbe obtained would add inanimportant way toour insights into thenatural behavior ofcomplex systems and would offer apotential foradvances in science andtechnology. ‘The prior artinmathematical analysis asseen intheliterature necessarily relies onsuch limiting procedures. Thus itmay well besaid that physics is usually perturbative theory andmathematics isessentially linear operator theory. Ofcourse there aresome methods ofsolving nonlinear equations, but not general methods. For example, clever transformation ofvariables sometimes results inalinear equation: however, thisrarely works. The objective ofthedecomposition method istomake possibie physically realistic solutions ofcomplex systems without theusual modelling and solution compromises toachieve wactability. Abonus isthat itessentially combines thefieldsofordinaryandpartialdifferentialequations.Thischapter will summarize the method and will briefly discuss applications and consequences foranalysis ahdcomputation Suppose wethink about physical systems described bynonlinear partial differential equations. Inthemore complicated problems, weordinarily must resort todiscretized methods and numerical computation. Anappropriate example isfluid flow and“computational fluid dynamics” (C.F.D.), anareeof TueDecouposmon MerHoo a intensive research inattempting todevelop coues forstudy oftransonic and hypersonic flow, Because ofthe symbiosis between such existing methodology andsupercomputers, aswell asthecomplexity, these methods arecomputationally intensive. Massive printouts aretheresult andfunctional dependences aredifficult tosee, Wehave aconstant demand forfaster computers, superconductivity, parallelism, etc., because ofthenecessity tocut down computation time. Thus acontinuous solution and considerably decreased computation isevidently adesirable goal. Closed-form analytical solutions areconsidered ideal when possible. However, they may necessitate changing theactual orreal-life problem toa more tractable mathematical problem. Except forasmall class ofequations in which clever transformations can result inlinear equations, itbecomes necessary toresort tolinearization orstatistical linearization techniques, or assumptions of“weak nonlinearity,” etc.What wegetthen issolution ofthe simpler mathematical problem. The resulting solution can deviate significantly from thesolution oftheactual problem; nonlinear systems canbe extremely sensitive tosmall changes. These small changes canoccur because ofinherent stochastic effects orcomputer errors; theresulting solutions (especially instrongly nonlinear equations) canshow violent, erratic (or “chaotic”) behavior. Ofcourse, itisclear thatconsiderable progress hasbeen made with thegenerally used procedures, and, inmany problems, these methods remain adequate. Thus, inproblems which areclose tolinear, or ‘where perturbation theory isadequate, excellent Solutions areobtained. Inmany frontier problems, however, wehave strong nonlinearities or stochasticity inparameters, sothat itbecomes important tofind anew approachandthatisoursubjecthere,»(0°eyVWebeginwiththe(deterministic) formFug(t)whereFisanonlinear ordinary differential operator with linear and nonlinear terms. We could represent thelineartermbyLuwhereListhelinearoperator. InthiscaseL must, becagilyinvertible which may notbethecase, ie,wemay have aontoease ‘Yanction andaconsequently difficult integration. Instead, we writethelineartermasLu+Ru wherewechooseLasthehighest-ordered derivative. NowLissimply ann-foldintegration forannthorderL,Theremainder ofthelinear’ opérator isR.(Incases where stochastic terms are present inthelinear operator, wecaninclude astochastic operator term ‘Ru.) ‘The nonlinear term isrepresented byNu.Thus Lu+Ru+Nu=g and we write 7 Lu=g-Ru-Nu fede LoLu=L"g-L"Ru-L"'Nu. 4, -of Gomi 4Forinitial-value problems weconveniently define L"'forL=d*/dt* asthe n-folddefiniteineggon‘operatorfrom0tot.FortheoperatorL=d’/dt’, forexample, wehavéL’'Lu =u—u(0)—tu’(0) andtherefore u=u(0)+w'(0)+L"'g-L"Ru-L'Nu For the same operator equation but now considering aboundary value problem, weletL”'beanindefinite integral andwrite u=A+Btforthefirst two terms and evaluate A,Bfrom thegiven conditions. The first three terms »ateidentified asuyintheassumed decomposition u=5”,u,.Finally, assuming Nuisanalytic, wewriteNu=S-", A,(up.t,...,u,) where theA, re arespecially generated (Adomian) polynomials forthespecificnonlinearity.£They depend onlyontheuptou,components andformarapidly convergent series. The A,aregiven as Ap=f(U) A,=u,(d/du,)f (Us) Az=ty(d/Att)f(up)+(u/21K{6°/dus )f(ue) A,=us(4/dup)f(¥) +1,43(6"/du5(uo)+(uj/31\4°/aus)E(u,) and can befound from theformule (for n>1) A,=Devan)" (us) Inthelinear case where f(u)=u, theA,reduce tou,.Otherwise A,= A,(Uot;, 5Up). Forf(u) =u,forexample, A,=u3, A,=2u,u,, A,=uj+2ugu,, Ay=2uyu; +2upus,... -Itistobbenotedthatinthisscheme, thesumofthesubscripts ineachtermoftA,aequalton.Thec(v,n)areproducts (orsumsofproducts) ofvcomponents ofuwhosesubscripts sum ton,divided bythefactorial ofthenumber ofrepeated subscripts. Thus c(1,3) “canonlybeus.c(2,3)isuyu,and(3,3)=(1/3!)u;.Foranonlinear equation in ‘Tne Decourosmon MerH00 ° (Pee .u,onemayexpress anygivenfunction f(u)intheA,byf=D>,AL Woriave {reviously pointed outthattheA,polynomials arenotunique, e.2., forflu)=u’, A,=uj, A;=2ugu,, A, =u;+2uyu;,.... ButA;could also be2u,u, +i, ie.,itcould include thefirstterm ofA;since uyandu,are known when usistobecalculated. ~InisnowestablishedthatthesumoftheseriesJ,A,forNuisequalto a)EPG ee ; DeesAs!.thésumof2generalized Teylorserieszboutu(x),thatJ",u,isequaltoa generalized Taylor series about thefunction us,andthat theseries terms Jy,e2proachzeroas1/(mn)!ifmistheorderofthehighestlineardifferentialIC operator.Sincetheseriesconverges (innorm)anddoessoveryrapidly,then=C/)termpartialsumg,=0"u,canserveas@practicalsolutionforsynthesis“ apddesign.Thelim9,=u.yrys.Pe CF CI" OM =CPLeamSs OO PE(jyBaer."OiiierConvenient algorithms havebeendeveloped forcomput’ aid7/2Toutidimensional functions aswellasforparticular functions ofinterest.Asanexample forsolution of.theDuffiig dquation, wewethenotationA,[f(u)]= ALWIL SUL ag OY1-1 - Ifwewritef(u)=D7,A,[f(u)] ormoresimplyf(u)=0™, A,andlet f(u)=u, wehaveu=D™, u,sincethenAy=uy, A,=u,,... .Thus we -fotss camsaywandf(u),i,thesolutionandanynonlinearity, arewritteninterms C7oftheA,,or,that wedothisforthenonlinearity andthink ofuassimply decomposed intd,components u,tobeevaluated suchthatthen-term approximation 9,=."u,approaches u=Jeasn+,Thesolutioncannowbewrittenas:C02 Lf ssh) YS.|OREw= usu,-L'RY u,-L"y A, = Fd Ef sothet u,=-L"Ru, -L"A, u,=-L"Ru,-LA, etc.Allcomponents aredeterminable since Aydepends only onup.A;depends onu,,u, etc. The practical solution will bethen-term epproximation or approximant tou,sometimes written @,{ul orsimply @,. 10 Curren 2 =Du, a limg,= usu Convergence hasbeen rigofously established byProfessor Yves Cherruaull [1}.Also further rigorous re-examination hasmost recently been done by Lionel Gabet {2}.Therapidity ofthisconvergence means thatfew terms are| required asshown inexamples, e.g., (1). BASISFORTHEEntchivensss OFDECOMPOSITION: (J2.=/Let'sconsider thephysical basisfortheSccu¥acy andrapid rateo! |convergence. TheinitialtermupisanoptimalfirstapproximatiGif containing i.GY?essentially allaPrigriinformation aboutthesystem. Thus,Wy=O+L"g| contains thegiven ‘input g(which isbounded inaphysical system) andthe| initial orboundary conditions included in which isthesolution ofL®=0. Furthermore. thefollowing termsconverge forbounded tas1/(mn)! where n| istheorder ofLandmis thenumber ofterms intheapproximant 0...Hence| even with very small m,the4,will contain most ofthesolution. Ofthe| ;,..following-derived terms, u,isparticularly simple, since Ao,thefirstofthe2’ polynomials representing thenonlinearity f(u),isflu,)whichofcourseisalso |.known, Thef(a)heednotevenbeanalytic [3]butcould bepiecewise- 1g! differentiable (Sobolev space) andrepresented byaseries ofanalytic] functions, Wedorequire bounded ©andg,which isphysically reasonable| and theu.terms must be(Lebesgue) integrable. For nthorder differential operators L,wemust have ann-fold differentiability/continuity requirement on| solutionsipordertoassureexistenceanduniqueness. Thedevelopment wasbasedorizconnpdsition inealeifable termsofthesolutiontobe’foumndrather| than anexpansion inaseries. Wewill seethat themethod works for initial- value orboundary-value problems andforlinear ornonlinear, ordinary or| partial differential equations andeven forstochastic systems, Weprove now] thatthe5A, isarearranged Taylor series about thisoptimal first} approximation 0,=up Toseethattheseries inA,polynomials forms ageneralized Taylor scries| about afunction (rather than apoint), wewrite THe Decosrosmow Merion " f(uy=5,A,=f{uy)+uf)(a9) +(u2/2(u,) +u,f(u,)+ust(0,) +uu,(u,)+(u3/3}¢(u,) + which canberearranged as fu)=f{u,)+(u,+,+...) (up)+[(u?/2!) +uu,+...JP(ug) + =tu.)+[(u-u,)/t]f(u,) =[(u-w,)* /24¢"(u,)-~ =Dla) Afro) AREFERENCE LIST OF THE ADOMIAN POLYNOMIALS: Ay=f(y) Ay=uf"(u) Az=ust(up)+(1/2)ujf(u,) Ay=ugf"(uy) +umf(ug)+(1/3!)u}t (uy)A,=ut(uy)+[(V2!)u+u,0,]f(u) +(1/2)ujuzt” (up)+(1/4!)u;(up) Ag=ust(uy)+(u,v,+0,4,]f(u) +{(0/2)uju3 +(Y2)uu,e(U4) +{1/3!)ufu,t (up)+(I/Stu;t (up) Ag=uct(U9) +[(I/2!)uj +u,u, +uu,f(y) +[(u3u; +uuu, +(Y/2)uju,]f(us) +[(y2)u(1/2)u; +(1/3))aju,je(U5) +(1/41)ufu,t (up)+(Y6!)uft (uy) Ay=upfl'(uy) +[Uyuy+gu,+uu,]f"(u) +[(y2)ezu, +u,(1/2)!u} +uju,u, +(V2)uFu,]0(u) +[us(3)u3 +(1/21)ujusu, +(1/3!)uru, J(u) 2 Cunoren2 +[(/3!)u}(1/21)u3 +(1/41)ufu, J(u) +(I/St)ufu.t (uy)+(1/7!)uzf(uy) Ag=£(u,)u, +£[(1/2)ui +uu,+u,u, +4,5] +£(ug)[u,(1/2!)uj +(1/21)uzu, +u,u,u, +uuu, +(1/2)uzu,] +£°%(us[(4t)u$+u,(Y/2)uzu, +(12!)uj(Y2)us +(Y/2)uFuzu, +(1/3)upus] +£(u9){(1/2!)u7(1/3!)u3 +(1/3))uzu,u, +(1/4!)uju,] +£(up)f(1/4!)uf (1/21)u3+(1/5!)uyu | +£(ug)(1/6t)ufu, +£(u,)(1/8!)up Ay=f"(up)uy +£(uy)[usus +usu,+UzU;+U,U5] +£°)(ug)[(0/3!)u} +wus, +(1/21)uzu, +u,(1/2uz su.uyu, =4,u,u, +(1/2!)uzu, | =f9(up)[(3)udu, +uju,(1/2!)u; +u,(/2)uzu, ~(1/2!)uzuu, +(1/2)ufuguy +(1/3!)u}u, | =£\(up)fu,(1/4!)u5 +(1/2!)u5(1/2!)uzu, +(1/3!)up(1/2!)uj +(1/3!)uju,u, +(1/4!)ufus] =£°(u,){(1/3})u3(1/32)u3 +(1/4))usu.u, ~(1/S))u‘u,} £7(u,)[(1/5!)u§(1/2)ud +(1/6!)ufu, | —£(u, O/Tuyu,+£(uy)(1/)uy Ayo=£1iyo+£"(up)[(1/2!)us +UU,+UU,+UU,+UU] +£(u,)[(/2)usu, +u,(1/2)uz +uju,u, +(1/2Juzu, u.uyu, =WyuU, +U,UsU,+(1/2))uzy,] =£(uy)[(1/21u3(1/2!)u5 +(1/3!)uzu, +u,(1/3!)u3 +u.u,u,u, +u,(1/2)uZ uy+(1/2)u; 1/2)uy =(1/21uF uyus+(1/2)uj uug+(1/3!u; u,) +£(uy){(1/S!)u$ +u,(1/3!)uzu, +(1/2!)uzu,(1/2!)u +(1/2!)u7(1/2!)uiu, +(1/3!)ujusu, Tus Decowrosmon Mer100 3 +(1/3!) ujusu, +(1/4!)usu,] +£°°(uo)[(1/21)us(1/48)u5 +(1/5!)u(1/2!)uu, +(1/4})u$(1/2!)u; +(1/41)uyuzu, +(1/S!)ufuy] £7(uy)f(1/4!)ut(1/3!)u} +(1/5!ufuguy +(1/6!)ufu,] +£(ug)(1/61)us(1/2:)u}+(1/7)ujuy | +£"(u, )(1/8!)ufu, +fu,)(1/10!)u)? EXAMPLE: Nu=u" Ag=uy A,=Sugu, A,=Suju, +10uyu; Ay=Sugu,+20upu,u, +10uzu? Ay=Sufu,+Sufu,+10uju +20u3u,u, +30ujuhu, So! Noticewhiteeachinaltermistheproductofmfactors.Eachtem ofA,hasfivefactors—the sumgfsuperscripts ism(or5inthiscase).Themecnisn.TheacteaeasanSeptisSnows andthé“suim ofsubscripts is4.Avery convenient check onthenumerical coefficients ineach term isthefollowing. Each coefficient ism!divided bythe product offactorials ofthesuperscripts foragiven term. Thus, thesecond termofA,(u*)hasthecoefficient 51/(3!\(1!)(1!) =20.ThelasttermofAyhas thecoefficient 5!/(2!)(2!)(1!) =30.Continuing withtheA,,foru’wehave A,=uj+Susu, +20u3u,u, +20uju,u, +20u;u,u, +30uzutu, +30uzu7u, Ag=Suju, +Sufu, +10u3u; +10uju} +20upu,u, +20u3u,u, +20uyupu, +30uju;u, +30u;uzu, +60u5u,u,U, A,=Sugu, +Sufu, +10u;u; +20u;u,u, +20uzu,u, +20uzu,u, +20u4u,u, +20u?ugu, +30uzulu, +30uzuu, +30uutu, +60uzu,u,u, +60uzu,u,u, M4 Charren2 Ay=Susu, +Suju, +Susu, +10u3u; +10uzu} +20u3u,u, +20uu,u, +20uzu,u, +20u;u,u, +20u;u,u, +30uzu,u5 +30uzuzu, +20uzu,u, +20uju,u, +20u;u,u, +30uju,u; +30u;uzu, +30upu;u, +30u;u;u, +60u;u,U,U, +60u,uzu,U, +60u;U,U,U, EXAMPLE: Nu=u? Ag=u A,=3uju, A,=3uju, +3ujuy Ay=u;+3ugu, +6u,u,u; A,=3uju, +3u7u, +3uzu, +6ugu,us Ag=3uju, +3u7u, +3uzu, +6u,u,u, +6u,u-u, A,=ul+3uiu, +3u7u, +3uju,+6upu,u,+6uu,u, +6u,U,U5 A,=Sulu, +3u2u, +3uju, +3uju, +6u,u,u, +6uyuzu, +6u,usU, A,=3uju, +3u7u, +3uju, +3uju, +3uzu, +6upuju, +6uyu,u, Gust, +6uyu,U, +6u,U,U, A,=u)+3ulu, +3uvu, +3uiu, +3uzu, +6u,u,u, +6u,u,u, =6ujUjU, +6upU,U, +6u,U,U, +6u,U,U, +6U,U,U, Avy=3ujUje =3u7u, +3uzu, +3uju, +3uzu. +3uru, =6u,U,U, —6u,U,U, +6UjUjU, +6U;u,U, +6u,U,U, +6u,U,U, =6u,u,U, +6u,u,U, EXAMPLE: Nu=u Ag=up A,=2uyu, A,=u?+2u,u, Ay=2uju; +2ugu A,=u}+2u,u, +2ugu, Ag=2uyu, +2u,u, +2ueu, EXAMPLE: N@=sin@ Tue Decouposmoy Merii00 15 A,=sin8, A,=8,cos, A,=-(67/2)sin@, +8,cos, A,=-(0;/6)cos@, -0,0,sind,+8,cos8, EXAMPLE: f(x)=sinh(x/2) at h(x,/2 A,=7H608(x,/2) 1 1s(ly (x Ae4,osh(x/2)+ Sai4)sim{2) 2 atAS=$seosh(x,/2)+(2) XXsinh(x,/2)+2i{3) cosh(x,/2) 1 1, 1) As$s,cosh(s,/2)+[ Eh+x6|(4) sinh(x9/2) oEate(Z)cost/2)+(2)stsinn2)22 re a2) ° EXAMPLE: f(u)=u"™ ,m>0 A= U5" A,=~muj"u, - A,=}m(m+ usu? ~mua, A,=—$m(m +1)(m +2)ug*u} +m(m+1)us'**u,u, -mu;'"*u, EXAMPLE: f(u)=u’ where 7isadecimal number. Ay=u Al=7p"4,A= uty; +£7(7— lustu Ageup,+7(7—Naguyu,+57(7~1)(y~2)usu} A= 10hag+(7—usu+uu)+47(7~1)(7-2)usuj, +hr(r- Wr 2)(7-3)uy“uy EXAMPLE: Consider thelinear (deterministic) ordinary differential equation du/dx? ~lofu=gwithu(1)=u(-1) =0.WriteL=d?/dx* andLu=g +kx?u. Operating with L"',wehave L"'Lu =L'g+L“kx*u. Then w=, +¢,x+ gx?/2+L keto LetSu, withuy=c,+0,x+gx7/2. Thenug.)=Lku, withm20. Thus usS(L'e Pu, =) u=DbFre,+P(t")Pe,x Ee)Eo Sew yen U=6,0,(x)+6.0.(x)+I(x) where o.(x)= Yk®x"°2*/(mp+ 2m-1)(mp+2m) cad 0,(x)= SYkex**?*"/(mp +2m)(mp+2m-1) (x)=¥(1/2)gk* x°°"7*"*/(mp +2m+1)(mp +2m+1) Since u(1) =u(-1)=0, wehave €,9,(1)+¢,9.(1)+T(1)=0 ¢,6(-1) +¢,0,(-1)+1(-1)= 0 Hence c,and c,aredetermined. Suppose that intheabove example, welet k=40,p=1,¢=2.Thus we consider theequation d*u/dx’ -40xu=2withu(-1)=u(1)=0."Thisisthe one-dimensional case oftheelliptic equation V7u=f(x,y,z)+k(x,y.z)u arising inproblems ofphysics andengineering. Here L=d?/dx* andwehave Lu= 2+40xu.This isarelatively stiff case because ofthelarge coefficient of Tue Décompasmon Merioo 17 wi,” , .u,and thenon-zero forcing function which yields anadditional Airy-like function. Operatingwin,L'yieldsu+A+Bx+L"(2)+L"(40xu).Let uy+A+BSE(2)£4$B£4?andtetu=Su,withthecomponents tobedetermined sothatthesumisu,Weidentify u,..=L“(40xu, ).Thenalt components canbedetermined, e.g..“2” r= u,=(20/3)Ax? +(10/3)Bx* +2x° u,=(80/9)Ax® +(200/63)Bx’ +(10/7)x* Ann-termapproximant @,=J")u,withn=12forx=0.2isgivenby 0.135649, forx=0.4isgiven by-0.113969, forx=0.6isgiven by 0.083321, forx=0.8isgiven by-0.050944, and forx=1.0is,of course, zero. These easily obtained results arecorrect toseven digits. Wesee that abetter solution isobtained and much more easily than byvariational methods. Thesolution isfound justaseasily fornonlinear versions without linearization. Cr" On‘AWALYTIC SIMULANTS OFDECOMPOSITION SOLUTIONS: Wenow introduce theconvenient concept of“simulants” tosolutions by decomposition. Them-term “approximant” @,tothesolution u,indicated by q{u},willmeanmtermsoftheconvergent series..™ ju,whichrepresents uinthedecomposition method.Ifwehaveanequation Tu=g(t)whereTisa general differential operator such as,forexample, & d 24a 8( +B geOFeGO+8O andwewriteg(t)=", gt"butonlyusemtermsoftheseries,wehavethe m-term approximant onle]= >&t° Thecorresponding solution oftheequation isthesimulant ofthesolution u, thus To,lu]=¢a[e] 18 Chaoren2 Analogous tothelimit mo of¢,[g]=g, thelimit asme of o,[u)=u. Possible stopping rules arise incomputerized calculation when thelast computed simulant ¢,.;{u) corresponds to@,(u) tothenumber ofdecimal places ofinterest tous. ‘We canalso conceive ofusing theentire series forg,forexample, writing thesum oftheinfinite series butusing asequence ofapproximants to parameters o,f, ..,inTandacorresponding sequence ofsimulants ¢,{u] as parametrized by9,[@] or¢,{4]. Insolving apartial differential equation by decomposition, wemay develop asequence ofsimulants ¢(u) forthesolution uby concurrently improving thelevel ofapproximation ofthecoefficients, or the given conditions, orthe input functions, For example in Log[u]+R,ca(u}= ¢a[e]whereg(t.x)= 7,D7,Beat/x™ wecan compute each o,(u) forgiven approximations ofg,and oftheinitial conditions, orfinally ofcoefficients such as aun=>¥aygt'x®of an We can also use theconcept ofsimulants with asymptotic decomposition which isdiscussed in[4).Consider theequation du/dx?-u=g(x)= >gx" u(0)=c, andu’(0)=c, Bytheasymptotic decomposition technique [5],theequation iswritten as usg(x)- d?u/dx? Thenu,=g(x)= 0,g.x"andu,=-(4°/dx")u,., form>1.Weusean approximant ofgor 9els}= >8.x" Thenthesimulant, oranalyticsimulant, ofuisthesolutionof Tne Decouposton MEnI00 1” @o,[ul ttl og{ul =0.)ae (u]=9.{8] or oul o,(u]=¢,(g]-alu]=¢q{8]-—F g,[u) isaserieswhichwewrite as)”,0”where oy"=0,[8] seg 8gin) OM s-—0") (neet ) $om=0,--“a'2Deal aalairead Itis straightforward enough that ifwedon’t useallofg,wehave only ful whichapproaches uinthelimitasm+ in0,=5°)u,.Inthesame equation u=g(x)—(d?/dx*)u & uy= £(x) ug =,19=(x) 2Gre wecanwrite use5ae af)=-(n+1)(n+2)a; sothat wed ad Saeed oF M-Sat a ae mS Fer} where a,=7, al.Therefore us3age a isthesolution forasymptotic decomposition. Wemake twoobservations: 20 Charen > 1)The method works very well fornonlinear equations where wesolve for thenonlinear term andexpressitintheabovepolynomials. 2)Ordinary differential equations with singular coefficients offer nospecial difficulty with asymptotic decomposition, e.g., (1=x)(4?/dx? )u+u=g(x) u=g(x)+(x-1)d*u/dx? up=g(x) REFERENCES 1. Y,Cherruault. Convergence ofAdomian's Method, Kybernetes, 18,(31-38) (1989). 2, L.Gabet, Equisse d'une théorie décompositionelle, Modélisation Mathématique et Analyse Numérigue, iapublication. 3. G.Adomian and R.Rach. Smooth Polynomial Approximations ofPiecewise- differentiable Functions, Appl. Math. Len. 2.(377-379) (1989). 4. G,Adomian, Nonlinear Stochastic Operator Equations, Academic Press (1986) 5. G,Adomian, AReview oftheDecomposition Method and Some Recent Results for Nonlinear Equations. Comp. andMath. with Applic, 21,(101-127) (1991). SUGGESTED READING 1, G.Adomian, R.E,Meyers,andR.Rach,AnEfficientMethodology forthePhysical Sciences, Kybernetes, 20, (24-34) (1991), 2. G..Adomian. Nonlinear Stochastic Differential Equations, J.Math, Anal. and Applic. $5, (441-452) (1976). 3. G,Adomian, Solution ofGeneral Linear and Nonlinear Stochastic Systems. in ModernTrendsirCybernetics andSysiems.}.Rose(ed.),(203-214) (1977). 4. G,Adomian andR.Rach, Linear andNonlinear Sebrbdinger Equations. Found. ofPhysics.21.(983-991)(1991), 5.N,BellomoarsR.Riganti,NonlinearStochastic SystemsinPhysicsandMechanics,World Scientific. (1987). 6. N.Bellomo and R.Monaco, AComparison between Adomian’s Decomposition Method and Perturbation Techniques forNonlinear Random Differential Equations, J Math. Anal. ard Applic.. 110, (1985). 7. N,Bellomo, R.Cafaro, and G.Rizzi. OntheMathematical Modelling ofPhysical Systems byOrdinary Differential Stochastic Equations, Math. andComput. inSimul4,(361-367)(2984), 8, N.Bellomo and D.Sarafyan, OnAdomian's Decomposition Method and Some Comparisons with Picard’s Iterative Scheme, J.Math, Anal. and Applic. 123. (389- 400) (1987). 9. R.Rach and A.Baghdasarian, OnApproximate Solution ofaNonlinearDifferential Equation, App, Math. Let.,3, (101-102) (1990) 10. -R-Rach. OntheAdomian Method andComparisons with Picard's Metbod,J.Math. ‘Anal. and Applic. 10,(139-159) (1984).11,R.Rac.AConvenient Computational FormfortheA,Polynomials, J.Math.Anal‘and Applic.. W2, (415-419) (1984). THe Decourosirion MerHoD 2 12,A.K.Sen,AnApplication oftheAdomian Decomposition MethodtotheTransientBehaviorofaModelBiochemical Reaction,J.Math.Anal,andApplic..131,(232- 245)(1988).13. ¥.Yang, Convergence oftheAdomian Method and anAlgorithm forAdomian Polynomials, submitted forpublication. 14. K.Abbaoui and Y.Cherruault, Convergence ofAdomian’s Method Applied to Differential Equations, Compur &Math. with Applic.,to appear. 15. B.K. Datta, Introduction toPartial Differential Equations, New Central Book AgencyLtd.,Calcutta(1993), CHAPTER 3 ‘THE DECOMPOSITION METHOD INSEVERAL DIMENSIONS ‘Mathematical physics deals with physical phenomena bymodelling the phenomena ofinterest, generally intheform ofnonlinear partial differential equations. Itthen requires aneffective analysis ofthemathematical model, such that the processes ofmodelling and ofanalysis yield results in accordance with observation and experiment. Bythis, wemean that the mathematical solution must conform tophysical reality, i.e.,totherealworld ofphysics. Therefore, wemust beable tosolve differential equations, in space andtime, which may benonlinear and often stochastic aswell, without theconcessions totactability which have been customary both ingraduate training and inresearch inphysics and mathematics. Nonlinear partial differential equations arevery difficult tosolve analytically, somethods such aslinearization, statistical linearization, perturbation, quasi-monochromatic approximations, white noise representation ofactual stochastic processes, etc. have been customary resorts. Exact solutions inclosed form arenota necessity. Infact, fortheworld ofphysics only asufficiently accurate solution matters, Allmodelling isapproximation, sofinding animproved general method ofanalysis ofmodels also contributes toallowing developmentofmoresophisticated modelling[1.2} Our objective inthischapter istoseehow tousethedecomposition method forpartial differential equations. (Inthenext chapter, wewill also introduce double decomposition which offers computational advantages fornonlinear ordinary differential equations and also fornonlinear partial differential equations.) These methods areapplicable inproblems ofinterest to theoretical physicists, applied mathematicians, engineers, and other disciplines andsuggest developments inpure mathematics. We now consider some generalizations forpartial differential equations. Just aswesolved forthelinear differential operator term Luand then operated onbothsides with L”',wecannowdothesame forhighest-ordered linear operator terms inallindependent variables. Ifwehave differentiation forexample, with respect toxandt,represented byL,u andL.u, weobtain equations foreither ofthese. We can operate oneach with theappropriate inverse. Webegin byconsidering some illuminating examples. Consider the example u/dt+du/ax+f(u)=0 with u(t=0)=12x andu(x=0)==1/. Forsimplicity assume f(u)=u*.Bydecomposition writing L,u=-(2/x)u-u’, thenwriting u=S0™, u,andrepresenting u? byu=D, A,derived forthespecific function, wehave Lu=-(/dny u,-¥ A, usu, -Li(#/any, u,-LiD A, Consequently, uy=u(x,0)=1/2x u,=-L;'(A/dx)uy -L7'Ay u,=-L)(8/ xu, -L'A, Substituting theA,(u*} andsumming, wehave ee eS ustt ete e.2x ae” 6x > u=to+tebey 2x 2x 4x which converges if(t/2x)<1tou=W(2x-1). IfwesolveforLyu,wehave Lyu=-(9/d)u-flu) ~ usuy-Li(a/d0>, ulDA, BOS uy=-Vt uy“L310/At),-LitAy=2x? oru=-(I/t)[1+2x/t+---] which converges neartheinitial condition if2x/t <1tous (2x0. Both operator equations yield distinct series which converge tothesame function with different convergence regions anddifferent convergence rates. Itisinteresting toobserve thatconvergence canbechanged bythechoice of theoperator equations. Inearlier work, thesolutions (called “partial Solutions”) ofeach operator equation were combined toensure useofallthe Pa Courrex § given conditions. We seethat partial differential equations aresolvable by fooking ateach dimension separately, andthus ourassertion holds about the connection between thefields ofordinary and partial differential equations. There is,ofcourse, much more tobesaid about this and we must leave further discussion tothegrowing literature, perhaps beginning with the introduction represented bythegiven references. Consider theequation u,—u,, +(d/dt)f(u) =0wheref(u(x,t) isanana- lytic function. LetL,represent °/3t? andletL,represent d°/ax?. We now write theequation intheform Lu-L,u=-(9/0t)f(u) Using thedecomposition method, wecansolve foreither linear term; thus, Lu=L,u-(a/at)f(u) Lyu=Lwu+ (d/at)f(u) Operating with theinverse operators, wehave v=o, +L)Lu-L;'(a/a1)f(u) us, +LLu+L;(9/at)f(u) where the,,, areevaluated from thegiven initial/boundary conditions. Generally, either canbeused togetasolution, sosolving apartial differential equation isanalogous tosolving anordinary differential equation with theL, operator inthefirst equation andtheL,operator inthesecond equation assuming therole oftheremainder operator Rinanordinary differential equation. The (6/4t)f(u) isanonlinearity handled asbefore. Thesolution depends ontheexplicit f(u) and thespecified conditions onthewave equation. Toillustrate theprocedure, wefirstconsider thecase with f(u)=0inorder toseethebasic procedure most clearly. Wehave, therefore, u,-u,, =0and wewill take asgiven conditions u(0, x)=0,u(t,0) =0,u(7/2,x) =sinx, uc.n?2) =sint. LetL,=0°/dt' andL,=d°/dx° andwrite theequation asLu=L,u. Following our procedure, wecan write either u=c, k,(x)+c,k,(x}t +LOLu orusc, k,(+e, k,(0x+ LiLu. ‘TueDecourosiriow MentonwScverasDIMEN 2s Define ®,=c,k,(x)+¢, k,(x)tand®,=c,k,(t)+c,k,(0)xtorewrite the above asu=0,+L;'Lu andv=, +LiLu. The first approximant ¢,isu, =,. The two-term approximant ¢,is uy+u, where u,=L;'L,u,. Applying thetconditions u(0,x)=0 and u(t/2,x) =sinxtotheone-term approximant 4,=u, =¢,ky(x)~c, k,(x)t we have o,k(x) =0 yk, (x) /2=sinx orc,=2/7andk,(x)=sinx. Thenextterm isu,=Lj'L,u, =L;'L,{c,t sinx},andwecontinue inthe same manner toobtain u;,Uy,...u, forsome n.Clearly, foranyn. u,=(L;'L,)°uy =c,(sinx)(-1)°E*" /(2n=D! Ifwewrite forthem-term approximant, wehave forthetwo cases: ©,=c,sinEee fon+D! Since ©,(1/2,x) =sin x k(x) =0 - casinx¥(/2)™" [es+Di=sinx Asm—-,c,-1. The sum approaches sintinthelimit, Hence our approximation becomes anexact solution u=sinxsint,(The same result can befound from theother operator equation.) Thus, inthis case, theseries issummed. Ingeneral, itisnotandwegeta series with aconvergence region inwhich numerical solutions stabilize quickly toasolution within therange ofaccuracy needed. Adding the nonlinear term does notchange this; theA,converge rapidly and the procedure amounts toageneralized Taylor expansion forthesolution about thefunction u,rather than about apoint. We call thesolution an approximation because itisusually notaclosed form solution; however, we Point out that allmodelling isapproximation, and aclosed form which 26 Coarren3 necessarily changes thephysical problem byemploying linearization isnot more desirable andis,infact, generally less desirable inthat theproblem has been changed toadifferent one. Recent work byY.Cherruault andL.Gabet onthemathematical framework hasprovided therigorous basis forthe decomposition method. The method isclearly useful tophysicists andother disciplines inwhich real problems must bemathematically modelled and solved. The method isalso adaptable tosystems ofnonlinear, stochastic, or coupled boundary conditions (asshown intheauthor's earlier books). The given conditions must beconsistent with thephysical problem being solved. Consider the same example u,-u,=0with 0Sx<mand t20, assuming now theconditions which yield aninteresting special case forthe methodology. u(x,0) =sinx w(0,t) =0 44(x,0)=0 wrt)=0 Decomposition results intheequations u=qk,(t)+c.k,(x+ LYLu u=c,k,(x)+e,k(xt=L Liu Theone-term approximant g,=u,inthefirst equation is uy=c)k(1)+ek.(OX Satisfying conditions onxwehave ¢yk,(t)=0 andc,k,(t)@=0.Henceu,= 0.The first equation clearly does notcontribute inthisspecial case; weneed only thesecond. Thus, Uy=Chk,(X)+C,k(XIE Applying conditions ont,c,k,(x)= sinxand¢,k,(x)t= 0.Hence u,=sin x u,=L)'L,u, =(-/2!)sin x u,=LjL,u,=(t°/4!)sin x o v=(1-0/2!+14/4tn...)sin x=sinxcost TeDecowposirion MEnHo0IWSEvERALDIMENSIONS 2 Wearedealing with amethodology forsolution ofphysical systems which have asolution, and weseek tofind this solution without changing the problem tomake ittractable. The conditions must beknown; otherwise, the model isnotcomplete. Ifthesolution isknown, buttheinitial conditions are not, they can befound bymathematical exploration and consequent verification. Finally, weconsider thegeneral form Lu=Lu+(a/a0)t(u) Lyu=L.u-(a/a¢)f(u) o u=ck,(t)+c.k,(t)x+Ly Lu+Ly(d/at)f(u) w=Ok,(x)+c,k,(x)t+Lj' Lu-L7'(9/dt) flu) Wenowletu=5.”,u,andf(u)=0"A,andnotethisisequivalent to letting uaswell asf(u)beequal to A,where theA,aregenerated for thespecific f(u). Iff(u) =uweobtain Ay=uy, Ay=uj,.., ie, ST,Aatuh=Du.Now usutLiL, >u,+Lbi(a/ay a, u=up +LL,>u,-L(a/ay a, Togofurther wemust have theconditions onu,Suppose wechoose u(0,1) =0 u(x,0) =£(%) ula.) =0 u,(%,0) =0 Satisfying theconditions, wehave c,k(t) =¢,k,(t) =0oru,=0.Therefore theequation involving L;’does notcontribute. Intheremaining equation we BetCk,(x)=f(x) andc,k,(x)t= 0.Hence, 2 Curren 3 uy=£(%) u,=L'L,uy -L7'(2/ ddA, u,=LiLu, -L7(@/ aA, ‘Thus,components ofuaredetermined andwecanwritep=So"ju,asan n-term approximation converging touasm+, Tocomplete theproblem, f(u)must beexplicitly given sothatwecangenerate theA,.Weseethatthe solution depends both onthespecified f(x)andonthegiven conditions. RESULTS AND POTENTIAL APPLICATIONS: The decomposition method provides asingle method for linear and nonlinear multidimensional problems andhasnow been applied toawide variety ofproblems inmany disciplines. One application ofthe decomposition method isinthedevelopment ofnumerical methods forthe solution ofnonlinear partial differential equations, Decomposition permits us tohave anessentially unique numeric method tailored individually foreach problem. Inapreliminary testofthisnotion, anumerical decomposition was performed onBurger's equation. Itwas found that thesame degree of accuracy could beachieved intwopercent ofthetime required tocompute a solution using Runge-Kutta procedures. Thereasons forthisarediscussed in 2). EXAMPLE: Consider thedissipative wave equation u,Uy, +(8/90)(u*) =g=-2sin?xsintcost with specitied conditions u(0,t) =u(x,t) =0andu(x,0) =sinx,u,(x,0) =0. We have up=k(x)+k,(t+Lg from theL,uequation anduseofthetwo-fold definite integration L;’and up=k,(+k,(x-Lg from theL,uequation andapplication ofthetwo-fold indefinite integration L;\.Either solution, which wehave called “apartial solution”, isalready correct: they areequal when thespatial boundary conditions depend ontand TueDecourosmron MeTHo0wvSEVERALDIMENSIONS 2» theinitial conditions depend onx.When conditions onone variable are independent oftheother variable, thepartial solutions areasymptotically equal. From thespecified conditions u(x,0) =sinxandu,(x,0) =0 k,(x) =sinx k,(x)=0 sothat u,=sinx—(sin* x)(1/2-1/4sin2t) ‘The n-term approximant 9,is =! Pa =LOL) uy-Li(a/a) DA, aS a where A,=Upty, Ay=Uyt ==Ugt;,... ‘Thecontribution oftheterm L”'g touoresults inseif-canceling terms, or “noise” terms. Hence, rather than calculating exactly, weobserve that ifwe useonlyus=sinx,wegetu,=(-t"/2!)sinx, u;=(t*/4l)sinx, etc.which appears tobeu=cos tsinx.Thus thesolution isu=costsinx+other terms, Wewrite u=costsinx+N andsubstitute intheequation foruand findthatN=0,ic.,theneglected terms areself-canceling andu=costsinx isthecorrect Solution. Itisoften useful tolook forpattems tominimize or avoid unnecessary work. - Tosummarize theprocedure, wecanwrite thetwooperator equations Lu=g+L,u-(a/at)f(u) Lu=-g+L,u+(d/at)f(u) Applying theoperators L;'tothefirstequation orL;'tothesecond, u=k,(x)+k,(x)t+Lj'g+L/'L,u-L7'(a/anf(u) u=k,(t)+k,(0x-Lig+Lj'Lu+Li(a/anf(u) Substituting u=)\~,u, andf(u)=7,A,,whereA,aredefinedfor f(u), wehave Up=k(x)+k,Qn+Li'g uy.Li Lyu,-Li'(a/d A, 30 Cuarren 3 o uy=k,(+k (Ox-Ljg uy=LL, +Ly(a/a)a, where, ifboth make acontribution, either can besolved toget ann-term approximation @,satisfying theconditions togettheapproximant tou,The partial solution asnoted carlier issufficient. The integration “constants” are evaluated from thespecific conditions, i.c.,each @,satisfies theappropriate conditions foranyn. The following example illustrates avoidance ofoften difficult integrations forsolution tosufficient accuracy inaparticular problem, The exact solution, which will then beused forasolution totwo orthree decimal places in physical problems, isoften unnecessary. Wecansometimes guess thesum of thedecomposition series inaclosed form andsometimes determine itby Euler, Padé, Shanks, ormodified Shanks acceleration techniques. However, whether weseethis final form ornot, theseries isthesolution weneed. EXAMPLE: u,uy+(9/dt)f(u)= g(x.t) Letg=2e" sinx—2e™ sinxcosx andf(u)=uu,. Theinitial/boundary conditions are: u(x.0) =sinx u,(x,0) =-sinx u(0,t)=u(z,t)=0 LetL,=0*/dt°andwritetheequation as Lu=g—(9/0t)f(u)+(0*/ax*)u (By thepartial solutions technique, weneed only theone operator equation the @*/dx? isweated like theRoperator inanordinary differential equation.) Operating withL;'defined asthetwo-fold integration from0totand writingu=7,u,andf(u)=D7,A,wheretheA,aregenerated for {(u)=uu,,weobtainthedecomposition components TueDecourosman Memtoo.wSevesas DiMexiaNs a uy=u(x,0) +4,(x,0)+Ljg ug.=-Li(9/ aA, +L(9? Jax")uy form>0.Then,since\™,u,isa(rapidly) converging series, thepartial sum9,=", u,isourapproximant tothesolution. Wecancalculate theabove terms Us,U,,...,Uq aSgiven. However, since we calculate approximants, wecansimplify theintegrations byapproximating g byafew terms ofitsdouble Maclaurin series representation. Thus wewill drop terms involving vandx’andhigher terms. Then etek 2 sinx=x © cosx =1-nr) sothat g=2]ree(x)-2{t-20+28)e{1-2 m2 Then L“'g=0totheassumed approximation, Hence uy=x=x - u,=Li(a/dt)A, +L)(3?/ax*)u, =xt°/2 Thus thetwo-term approximation is @,=u,tu,=x-tx+xe/2 =(I-E+0/2)x=e'sinx Although wecancalculate more terms using u,., form>0,substitution verifies thatu=e“sinxisalready thecorrect solution.Ifweneedtorecognize theexactsolution,wecancarrytheseriesforgtoa higher approximation toseetheclear convergence toe“'sinx. Once we guess thesolution may bee“sinx, wecan verify itbysubstitution, or substitute e~'sinx+NandshowthatN=0. EQUALITY OFPARTIAL SOLUTIONS: Insolving linear ornonlinear partial differential equations by decomposition, wecansolve separately fortheterm involving either ofthe highest-ordered linear operators” andapply theappropriate inverse operator toeach equation. Each equation issolved forann-term approximation. The solutions ofthe individual equations (¢.g., forL,u, Lu,Lu, orL,uinafour- dimensional problem) have been called “partial solutions” inearlier work. ‘The reason was that they were tobecombined into ageneral solution using alltheconditions. However, ithas now been shown [4]that inthegeneral case, thepartial solutions areequal andeach isthesolution. This permits a simpler solution which isnotessentially different from solution ofan ordinary differential equation, The other operators, ifwesolve forLu, for example, simply become theRoperator inLu+Ru+Nu= g.The procedure isnow asingle procedure for linear ornonlinear ordinary orpartial differential equations. (When theupterm inoneoperator equation iszero, thatequation does notcontribute tothesolution andthecomplete solution is, obtained from theremaining equation orequations.) Wewill show thatthe partial solutions from each equation lead tothesame solution (and explain why theonepartial solution above does notcontribute tothesolution). Consider theequation L,u+L,u+Nu=0 with L,=0°/dx" and L,=4°/4y*, although nolimitation isimplied andNuisananalytic term accurately representable bytheA,,polynomials, Wechoose conditions: ula,.y)= ay) u(x,b,)=Bx) wla;.y)= @(y) u(x,b.)= B(x) Solving forthelinear operator terms Lyu=-L,u-Nu Lyu=-Lu-Nu ‘Using the“x-solution”, wehave LyLyu=-LjLju-LyNu *Purely nonlinear equations orequations inwhich thebighest-ordered operator 1snonlinear require further constaeration [3] TueDécouposrion MemonSevensDiuexsions Fn where L?isanindefinite two-fold integration andL;'()=j [()dxdx+, where®,=,(y)+xé,(y). The&(y)and&(y)arematching coefficients to theboundary conditions. Hence, L'Lu=u-®, and®,=4,(y)+xG(y), where ¢,and&,aretheintegration “constants”. Wenowhave u=0,-L{Lu-L{Nu Welet u=Yu, andNu="A, where theA,aredefined specifically forNu. The u,term isnormally taken as©,(or®,Lig when there isan inhomogeneous term). We can take asomewhat different approach (double decomposition), discussed inChapter 4,anddecompose ®,aswell. Inthatcase, wewrite Pi =Soa Xin and Uy=O, 5 ‘Then instead ofthefollowing components being given by ug.=-LiLu, -LJA, form21 wewould have u,=®,,-LLyu,- LIA, u;=; -LyLyu,- LA, u,=0,,-LiLyu, -L'A, uy=®,_Libya. LiAa.) Theboundaryconditions Prov(BY)=Cy) Pour(@ss¥)=Oy) determine &,.,and&,,,.Then u,=,, u,=O,“LAL, O,9LyAg M Cuarren 3 u=, .-L{L,0,, +LjL, 7, +LYL,LYA,- LyA, u,=,,- LIL,®,, +(LYL,P',, -(LAL, ~(LIL, LIA,+LL,LZA, -LA, uy=CLL, ®,..- DLL PL A, Now vasy= DYYELL. Y-LIL ELA, at ae which isthesolution totheequation inx.We can proceed inthesame manner with theyequation; however, wereturn totheordinary orregular decomposition forclarity. The additional decomposition isofnoadvantage forinitial-value problems butspeeds upconvergence inboundary-value problems bygiving usresults foru,,u,,...that areobtained bycorrecting constants ofintegration asweproceed. sothat Wecanthen useacorrected initial value without more matching toconditions. From thexpartial solution. Up=®,=Ea(y)+xGi(y) uy,=-LyLu,- LA, u,=(LVL, uy+ILZL LA, -LA. uy={LDL Puy-LL PLYAg=ILL,JLyA,“LYAS From theypartialsolution uu,=@,=Mp(X)+y7},(x) u,=LyL,u,- LA u,<1L,Fu+L IA:-LA, Themthapproximant @,,=""'u,ineachcaseabove, Theintegration constants aredetermined bysatisfying thegiven conditions bysolution ofthe matrixequations 1af] _fair Ja,JL&JLaaty). 1bi]fm]_ [Boo 1bsJinJ[B,00, todetermine 54,§.MusMh Thelimit asm— ©of@,,forthexequation andtheyequation arerespec- tively thexpartial solution andypartial solution andareidentically equal; either istheactual solution which satisfies thedifferential equation uniquely forthegiven initia/boundary conditions. REMARKS: Suppose weconsider apartial differential equation whose solu- tion isthesurface u(x,t) ina Cartesian system. We write this intheform Lu+L,u+Ru +Nu=g.Theintersections with theu,xplane isu(x,0) =f(x). Astincreases fromthisinitialvalue,thesurfaceuisgenerated. Similarly, the intersections with theu,tplane isu(0,t) =g(t). Asxincreases, thesurface is generated. The partial solutions represent these two possibilities, ic., we can determine ueither bystarting from f(x) andusing thetequations (Lu = g—L,u-Ru-Nu) orstarting from g(t) and using thexequation (L.u = g—Lu—Ru—Nu) andtheappropriate inversions foreach. ea Consider, asanexample, thesimple heat flow equation u,=u,,, given thatu(x,0) =sin(x/é) andu(0,t) =u(é,t) =0.The solution is 36 Cuaron § oFsin(ex/l) The equation intisLu=Lu. Applying theL;’operator, weget u=u(x,0) +LL,57,uy uy=sin(x/0) u,=LiL,uy=(27t/é7)sin(7x/é) u,=(2*/¢*)sin(wx/2) usYu,=e" sin(ax/e) which isthe complete solution usually obtained more easily than the textbook solutions ofthisproblem. Thexequation isL,u=L,u. .Applying Ly w=k()+xk,O+LL, Duy Wesee uy=k;(t)+xk,(t) =0whichmeansallfollowing components must bezero sothis equation, aspreviously stated, makes nocontribution. Here, thexconditions (boundary conditions) u(0,t) andu(Z.t) donotdepend ont. Hence thepartial solutions areasymptotically equal—they both arezero at (oe. Use ofthepartial solutions technique ascompared with theauthor's earlier treatments ofpartial differential equations [4]leads tosubstantially decreased computation andminimization ofextraneous noise {5}.Also wenote that the convergence region canbechanged bythechoice oftheoperator equation. Since thepartial solutions areequal, weneed solve only one operator equation. (Exceptions occurwhentheuptermiszeroinoneofthe equations ortheinitial/boundary conditions foroneoperator equation donotinvolve remaining variables.) The remaining highest-ordered linear differential operators cannow betreated liketheremainder operator R.Thus ordinary or partial differential equations aresolved byasingle method. The decision as towhich operator equation tosolveinamultidimensional problemismade The DecouposimionMETHODWSEVERALDIMENSIONS 7 onthebasis ofthebest known conditions and possibly also onthebasis of theoperator oflowest order tominimize integrations. Tomake theprocedure asclear aspossible, weconsider first thecase where Nu=0, ice.,alinear partial differential equation inR*, Lu+Lu+Ru=g where L,=4°/dx’ andL,=0'/dy’ with the boundary conditions specified byboundary-operator equations Buu}..,=Bly) Baul..2,=Bay) Buu].,=71(%) Bau]yan.720%) Solving forL,uwehave L,u=g —L,u ~Ru andoperating with L?wehave u=0,+Lig-LiLu-L7Ru where ®,satisfies L,®,=0. The inverse operator Lj!isatwo-fold (indefinite) integration operator since L,isasecond-order differential operator. The “constants ofintegration” areadded foreach indefinite integration. **(This makes notation consistent withdecomposition solution of initial-value problems whereforL,=9/dtwedefineL;'(]={i[Jét, andfor L,=07/dt", wehave atwo-fold definite integration fromzerotot.)Now thedecomposition u=D°~,u,yields u=,+Lig-LiL, Yu,-L:RDw, where weidentify u,=, +L “*For linear ordinary differential equations, butnotpartial differential equations. wecan View L!asapuretwo-fold integration operator notinvolving constants andsimply add the©,forageneralsolution. 38 Cuurres 3 astheinitial term ofthedecomposition. Since L?isatwo-fold integration, ,=coy) +xe\(y) anduy=cy(y) +xc,(y) +Lig. Hence u=u-LyL, Yu,-LiRD Then form >0: m Up. =“LyLju, -LZRu, forcomponents after us.Consequently, allcomponents ofthedecomposition areidentified and calculable. Wecan now form successive approximants =Doueasnincreases whichwematchtotheboundary conditions. Thus 9;=Ugy := +Uy,P= G2+Us,Serve asapproximate solutions of increasing accuracy asne and must, ofcourse, satisfy theboundary conditions. Beginning with g,=u,=c,(y)+xc,(y)+L;'g, weevaluate c, andc,from theboundary conditions Biro, =BY) BsPfans, =BY) Thus @,isnow determined. Since u,org,isnow completely known, we form u,=-LL,u,-L Ru, Then 9.= 9, uy, which must satisfy theboundary conditions. Continuing toanacceptable approximation g,,wemust match theconditions ateach value ofnfora satisfactory solution asdecided either bysubstitution orbyastabilized numerical answer toadesired number ofdecimals. Forthespecialcaseofalinearordinarydifferential equation, wehavethe simpler alternative ofusing theunevaluated u,togetu,,simply carrying alongtheconstants inu,andcontinuing inthiswaytosomeg,.Thus,inthis case, only one evaluation oftheconstants isnecessary. (For nonlinear cases orforpartial differential equations, thesimpler procedure 1snotgenerally possible.) Tomake this clear, consider some examples: Tue DecoMposimion MemowSeverusDiMENsows 9 @u/dx? -40xu=2 u(-1) =ul) =0 Write Lu=2+40xu o usc, +¢,x+L"(2)+40L "xu, Wecanidentify~ Uy=6,+6,x+L(2)= 6+0,x=x* Now instead ofevaluating theconstants atthisstage, wewrite u,=40L™ xu,-(20/3)¢,x? +(10/3)e,x* +2x° u,=40L"'xu, ~(80/9)e,x" +(200/63)e,x" +(10/7)x* If,forexample, @,issufficient, = C,+C,xX+x7 +(20/3)¢,x” +(10/3)e,x* +2x° +(80/9)¢,x° +(200/63)c,x” +(10/7)x* Imposing the boundary conditions at-1, and 1on gwe have 9,2 =9(1)=0or 149/9 473/63) (¢,)_(-31/7 2919-53163) \c,) \-3/7 determining c,andc,andtherefore @,inasingle evaluation, (By g,.this yielded seven-digit accuracy.) Another example is.d”y/dx? +2xdy/dx =0with y(0)=0andy(a)=1.The solution isy(x) =(erfx)/(erfa)or =(0/3) +(0°/10)=(x7/42) + a=(a?/3)+(a?/10)—(a" /42)+--- Write Ly=-2x(dldx)y or 0 Charen 3 Ifwesatisfy theconditions with g,=y, wehave y,=x/a asour first approximant. Ifwecontinue tosome gy,andevaluate only then, wehave yy=—2L'x(d/dx)yy =-2L'x(d/dx)(C, +cx)=-C)x7/3 Yz=-2L"'x(d/dx)y, =-2L'x(d/dx)(- 64x"/3)=—c,x°10 Ifwestop at B= Yor MtY2 x=,+€,x— C,x°/3+,x5/10 andnow satisfy theconditions wehave ya222/39 10)a-(a°/3)+(a*/10) which is(erfx)/(erf a)tothisapproximation which can, ofcourse, be carried asfar aswe like. Forlinear ordinary differential equations, both procedures will work, i.¢., wecanusetheevaluated u,togetu,,addittotheunevaluated u,toget9, then satisfy theconditions attheboundaries, orcarry theconstants along as intheexamples, Thelastprocedure ismostconvenient becauseofthe single evaluation: thefirst ismore general since itapplies tononlinear ordinary differential equations andlinear ornonlinear partial differential equations as well [6.7]. EVALUATION OFCOEFFICIENTS FOR ALINEAR PARTIAL DIFFERENTIAL EQUATION: Uy—Uy =0 forOSx£7/2,0Sy<A/2withtheconditions givenas u(0,y)=u(x,0)=0 u(r/2y)=sin y u(x,/2) =sin x LetL,=0°/ax? andL,=d°/dy’ towrite L,u=L,u. Ifweapply inver- siontotheL,operator, wehave u=k,(y)~ xk.(y)+L'Lyu. ‘TueDecowrosirion METHODIVSEVERALDIMENSIONS 4 Now ©,=k,(y)+xky(y). Hence =, +L/L,u. Theone-term approx- imant isg,=u,=,. Atwo-term approximant is9,=9,+u, and u,=LiL,u,. Thexconditions areu(0,y) =0andu(7/2,y) =siny.Applying these conditions tok,(y)+xk,(y) weseethat k,=Oandk,=(2/z)sin y. Thus, iftheone-term approximant g,were sufficient, the“solution” would be9,=(2/m)xsiny. ‘Thenexttermisu,=L{Lju, =L7L,{(2/z)x siny].Theng,=u,+u,is givenby9,=k,+xk,~(2/z)(x/3!)sin y.Because ofthe condition atx=0, wehave k,=0. From thecondition onxat%/2, (712)ky(y)~ (2/)((7/2)/3!)siny=siny k,(y) =(2/a=a/l2)sin y hence 9,=(2/t-7/12)x siny The first coefficient was (2/7)=0.637.Thesecond(from9,)is0.899.Asn=,thecoefficient approaches 1.0sothatu=xsinyisthesolution. Notice that ifwetrytocarry along theconstants ofintegration, k,andk,, tosomeg,anddoasingleevaluation fordetermination oftheconstants, we have u,=LL,u, =LyL,[k,(y)+xk,(y)] which wecannot carry out;we must usetheevaluated preceding terms rather than asingle evaluation at¢. Wehave used only theoneoperator equation; thesame results areobtained from either. Let usconsider amore general form forthelinear partial differential equation Lu+Lju+Ru=g where L,=0°/dx* andL,=d'/dy* with conditions specified byB,u(x)],.., =B,(y)andB,u(x)|,.4, =B,(y). Solving forL,u andapplying theinverse operator, wehave u=O+Lig-L{Lu-L{Ru where ©=c,(y)+xc,(y)isthesolutionofLo=0. Letu=7,u,andidentify theinitialtermasu,=¢,(y)+ xc,(y)+L;g. Now form>0,theremaining components areidentified 2 Cuarren 3 Wenowformsuccessive approximants p,=\*"u,, which wematch to theboundary conditions, Thus y= Uy,@=, +Uys y=, +Ug, =Serve asapproximate solutions ofincreasing accuracy asnapproaches infinity and must satisfy theboundary conditions. Beginning with 9,=Uy=Oy)+Xe)+L'g weuseB,g|,, =B(y)andBy,=By(y)todeterminec\(y)andc,(y)so that g,iscompletely determined, Since u,isnow known, wecan form u,=-L;'L,u,-LyRu,. Then g,=¢,+u, which must alsosatisfy the boundary conditions. Continuing tosome g,wematch theconditions fora sufficient approximation, Thus carrying along constants to@,forasingle evaluation doesn't work except forlinear ordinary differential equations. For linear partial differential equations, wemust use thealready evaluated preceding terms and can dosoalso fornonlinear ordinary differential equations. COEFFICIENT-GENERATING ALGORITHMS FOR SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS INSERIES FORM: Let's consider amodel system intheform eu au 28, us ple aot oeUtBGs=alts) assuming conditions givenintheformu(0.x)=o(x)anddu/at(0,x) =n(x). Write a(tix)=>Ygast™s®ced a(x)=3 9x" m n(x)= >nx" We note that Tue Decouposmo MemtovSevenDIMENSIONS ” (x)= >,g,(x)t® where, 2a()= >Baak® Defining L=4?/at* andL"asatwo-fold definite integration from 0tot, wwe can write Lu+au+B(d?/dx")u=a(t,x) o Lu=g(t,x)-@u-B(a*/dx")u Operating with L*,wehave usu, -L'au-Lp(#"/ax")u where u,=u(0,x)+ Fx) +L'g(tx) o gals"1p=O(x) + t7(x =vena) ine)+Bema d) which we will write as uy=Yal(x)e where os a(x)Fax? Thus thecoefficients are af)(x)=0(x) a(x)=n(x) af?) (x)=fal)22)=Nan) Usingdecomposition u=x2. Us, uu, —L'au-L"'B(d?/dx")u usuy-L'a Siuy-L1B(a*/ax")>) u,Land me0 sothat form =0, bo=Dali(xye and for m> 0,= uy=Lieu, ,-LB(#/8x*)ug., Sincewecanalsowrite v= DDaehtx” we noteareas ann=oy aan, a, <=—Se= oes" (m=1)(m~2) The next component u,is u,=-L'au, -L'B(07/9x*)u, Since (°/8x*)uy =(8*/Ox*)S a(x)je® =(#F/ax)d Yalieext et eo = Liw= ins 2hall), ex" ms Thus ee almaweed 2msnimed) pS Fg getge (2+DIn+2)>xSowa Umiym*2) which we write as Twe Decourosrow MerioowSeversDIMensions 45 as where at,=abl=Bln+Dint 2aaa (m+i(m+2) Proceeding inthesame way, write u,=-Lau,-Lp(F*/ax*)u, (@/ax)u, =D Yaly(n+l) (n+2)e2x" an ah¥alltetTEs BS(m+Ima) By Fath GAN =2t8"*x*ByzSeeTema) which isnow rewritten as. weed Sahin where meee af)=2@a@s —Bln(n+2)ao.e se (m+3)(m+4) Continuing, wecalculate us,us,...andseethatwecanwriteforthe4thcomponent ofu w-FsFleet(yoo) withas ale)=2eaES-BlnNin+2),sa (m+2u-1(m+2p) asacoefficient-generating algorithm. Thus forp=0 ate=e af=, a), =——Sas_srt8(m+1)(m+2) 4s Cnarren3 Thenthesolution isu=)™, u,.Consequently Sey Falee isthesolution with: =F EF aes asthe ¥'th approximant tothesolution which becomes anincreasingly accurate representation ofuasVincreases. MIXED DERIVATIVES: Consider theequation u,,=—u given theconditions u(x,0) =e*andu(0.y) =e.Let L,=a/ax L,=aay Then Ly')=f'Qdx andLQ fOdy. Inoperator form, wehave L,L,u =~u. Operating withL;'wehave LIL) =u Lyu-L,u(0,y)=-Lj'u Lu=L,u(0.y)-Lyu Operating nowwithL;! LyLu=L}'L,u@y)-LS Lu u-u(x,0)= u(0.y)- u(0,0)- LyLyu u=u(x,0)+u(0,y)-u(0,0)- LELyu Let u,=u(x,0)+u(0,y)— u(0.0) u,=et+e?-1 uy=-LI Liu, u,=-LILyu, TeDecourosirion MeT#70:NSevenDIMENSIONS ” Hence u=7,wor w=DCL Lu, Eo Since ug=(-LLY)%(e' +e?-D we have -pyey aieney (yy a,=(GL) 24-uL)y pie2 cyte ot ee CyOry ey OYm!%(m+y)!mi(m+4)! Because u=7,uy. =fore ot esey] vey(O*yey OY 5{m!%(m+y)!ma(m+py!) isthesolution. However, wecanrearrange theterms togetasimpler repre. sentation using staggered summation: Sem yy usoy>p(m=p)! pt ZS1S (mM)oy where (")-mi nw) (m= pint Werecognize thebinomial expansion of(x-y)"andwrite w=}to-yyeAm which is,ofcourse, theexponential series of(x-y) sothat u=e'? which is the same result inaconvenient form. “6 Conrrex 3 REMARK: Ifwewriteg;=u,+u,,wecanrecognize thefirstsixtermsof (+x+x?/2)-(l-y+y?/2)= ete. Write u=e%e? +N,substitute intothe original equation andseethatNmust vanish inthelimit. EXERCISE: u,=u,+u,~u withu(x,0)=e* andu(0,y)=e". (The solu- tion isu=e") EXERCISE: u,,=[4xy/(1+x’y')]u withu(x,0)= u(,y)=1.(Thesolution isu=I+x’y?) Ageneralization tou,+k(x,y)u= g(x,y) with u(0.y) =&(y) and u(x,0)=7(x) canalsobeconsidered using power series expansions ofthefunctions toseveral terms. MODIFIED DECOMPOSITION SOLUTION: u,,=u,=u,—uwith u(x,0) =e"andu(0,y) =e", Let w=DD arty? Then => DY(meday.. xy" w=>Ymtday,.x*y" uy=D Y(m=YO=Yax?y ad tom Ztox wOy)= >ay® Wenotethatuix,0)= 07,x*/m! andu0.y)=07, (-y)*/n!. Sub- stituting intheequation, Tue Decowrosriow MentonSevenDENons ~ YY m+dea axy¥= >Y(m+Nazi.x"y” +L LD@tnaxy- >Yaney Equating like powers (m+Nagg+ (0+Nage1 Bon BagageyBt = (m+in+1) Ago=Vm! a,=(-D*/a! Wecannow compute atable ofcoefficients inaconvenient triangular form aon Ay ays Ang Ay Ao Le whichisgivenas: re 1 1-1 1 1 2.4 21 2 2 foot ha 6 2 2 6 to ft aot m4 6 4 6 andbyinduction, a=mint Therefore Se ae us aga X= cry -F FSO _F 8FOr Consequently, u=e'?, From thetable ofcoefficients, weseethat 30 Cuaron 3 see oy 21s(m.. usSo et xB-8(— LXwoo 2hmylaPO Since oF (Mee yex-y)P= =(x-y)>(tscy FyGay oyod ese ADDENDUM: From thetable ofcoefficients intriangular form, wehave a.“min! Therefore bysubstitution, (m+1Day.,.++ Dag. =fomP+pyOE}{(m+n! mi(n~10"! =o 0")4min! mintJ Consequently wecanderive therecurrence relation bysubstitution: Mae Agee)=— =)"ne (n+1) a= /m! a,=(-1)"/n! sothat u=y Dasty! ae GENERALIZATION OF THE A, POLYNOMIALS TO FUNCTIONS OF SEVERAL VARIABLES: Inapplying thedecomposition method tononlinear differential equations arising inphysical problems, wemay encounter nonlinearities involving several variables [8]. We now generalize thealgorithm forA,forf(u) to analytic functions ofseveral variables such asf(u.v), where f(u.v) isnot tactorable into {,(u)f(v). (The latter case, ofcourse, issolvable asa“product TieDecomrosimon MEriootSeveRaLDIMENSIONS si nonlinearity” bydeveloping the A,foreach factor and obtaining their product.) Examples appear in(2].Our objective istoextend theclass of solvable systems. Intheuseofthemethod,thesolutionofadifferential orpartialdifferential equationiswrittenu=.~, u,andflu)=",A,(up,uy,.... ¥,)whereuy isafunction involving initial/boundary conditions, theforcing function, and anintegral operator. This amounts totheassumption thatthesolution and functions ofthe solution areexpanded intheA,polynomials since A, reduces tou,forf(u)=u.Fordevelopment ofanalgorithm fortheA,,itis convenient toassume parametrized forms oftheuand {(u). The following expressions have been given bytheauthor (2}as: Ag=(1/n(6/d2*)fluCA)ano wo orsimply A,=(I/n!)D*f],., where D*=d°/a2? and (2/02? )u(A)|sno =ny TheD*ftermforn>0canbewrittenasasumfromv=1tonof terms "f/du" with coefficients which arepolynomials inthed’wdA”. Thus, D'f=(df/du)(du/da) D'f= (d*t/du*)(du/da)* +(df/du)(d*wd?) D'f=(°f/du*\(du/da)? +3(d?t/du*\(du/da Kwa?) + +(df/du)(d’wdi?) ‘The result forA,can finally begiven inavery convenient form which we have referred toasRach’s Rule, A=, (v,0)f(up) @ Here (up) means thevth derivative off(u) atu=up andthe I/n! is absorbed inthec(v,n). The first index ofc(v,n) progresses from 1ton along with theorder ofthederivative. The second index istheorder ofthe polynomial. 32 Cuaron 3 TheA,isafunction ofto,Uy... iL€.,ofthecomponents ofthesolutionwinthedecomposition. Thec(v,n)areproducts (orsumsofproducts) ofv components ofuwhose subscripts sum uptonwith theresult divided by the factorial ofthe numberofrepeated subscripts. Forexample ¢(1,3)canonlybeu;(asinglesubscript whichmustbe3).¢(2,3)canonlybeu,1, (two subscripts adding to3).(3,3)=(1/3!)u}. (2,6)hastwosubscripts adding to6forwhich wehave three possibilities forujusing (2,4), (1,5) and(3,3), Hence c(2,6)=ust+Ujus+(1/2!u3. Theresultis Ao= fluo) A,=u,(d/dup}f(Ue) Az=Ua(G/duflte) +(Uj/2!9(0"/du 5flo) Ay=Us(@/dup)f(uo) +uu;67/dufluo)+(w}/39(4°/du§ flue) ANALYTIC FUNCTION OFTWO VARIABLES f(u.): Proceeding analogously tothecase ofonevariable, D'f= (dffduydu/da) +(f/dv\(dv/d2) D*f=("wad )\df/du) +(d?v/dA*)idf/dv'+ (du/dA)*(G"f/du’) +2(du/d.)(dv/dAXG*/dudv)+ (dv/d/)*(67#d7) D'f= (d'wei'\(df/du) +(6v/a2\f/dv) +3(du/d2)(d?wd2*\6"t/du*) +3(dv/d2\(d*wd2?\(6"f/ dud) +3du/d Ad? v/d2?Xd"f/dud “)*3(dv/dAXd? vida? \d"f/dy" ) +3dv/dA\idu/d2)°@fldvdur +3(du/dAy(dv/day'(d*f/dudv?) +(du/da)*@'Eldu’) +(dv/dA)*@?E/dv") Déf=(d*w/d2*)(df/du) +(d*vidA‘\(AE dv)+6(du/d2)*(d*wd22\(a°f/du?)* 6(du/d2)"(d*v/d 7X0"f/du'dv) +12(du/dAy(dv/ddy(d*w/dA?Xa"f/ du"dv) +12(du/d2)(dv/dy(d?v/d27\d°f/dudv*) +6(dv/d4)N*w/dA2\(6°E/dudv) +6(dv/d2)#(6*v/dA7 \d°£dv*) +3id*uldA?"(G"f/du’) +4(du/d2\id*wd4'\d*f/du*) Tne Decouposnon METHOD INSEVERAL DIMENSIONS 33 +Mdv/dAy(d? dd?\(d?t/dvdu) +6(d*v/dA?\d*wda2X edfidudv) +4(du/dA\d?v/dA? YXd"£/dudv) +4(dv/dAXa WidXPtd?) +3(d?vidA?Ud"flav’)+A(dv/dA)(du/dAy "tlavdu>) +6(dv/dA)"(du/dA)'(G*f/dvédut +4(du/dA)(dv/dA)s(dst/dudvs) +(du/dd)*(d*t/du') +(dv/da)"(a"t/av') The A,for f(u) are written A,(f(u)}. Generalizing toA,(f(u.v)} or Ag(f(u(),v(2))} weintroduce thenotation 10), Wan =6,.(00.%9)ao ,V(A))fano=Lar(YorVo, Proceeding analogously tothec(v,n) andf(up) forf(u), wecannow write c(u.v,n) and £*”,orf,,(Uy,Vo) forafunction f(u,v). Ay=(1,0,Df;p(to.¥»)+(0.1.1, (to.¥o) =¢(1,0,1)df/duy +c(0,1,1)df/dv, Comparing with D'f weseethat c(1,0,1) =du/da. which must beevaluated aA=0. Since u=7,Hu,,du/dajs.,=u,. Hence Ay=undffduy +v,df/dvo Ai=Urfia +Vifor Proceeding inthesame way wecanlisttheA,(f(u,v)) Ao=fao AL=Uifis +Vifar Al=Uafiy +Var +(UT/2Dfoo+UrVif,+(VF/2)Fos Ay=Usfig +Votan +UsUafeo +[U.Va+UaVyHE)+ViVafoa+(UI/3Dfap+(UT/2)Vifar +(Uivj/2Dfi2 +(V}/3Dfas AgUshio+Valo)+(03/2)+susrot(Wi¥s+tvs+Usd+[(v3/2! +vyvo]foa +(u7/2Duafyy+f(u 7/2)va+ witsys ay +[Wayvp+UV7/22+(VvF/2DVafas+ (Uf/4DFoo+(UI/3DV fay +(UPD PDfas +UVM s+(v{/4DFos 5 Charron 3 Pethaps more conveniently wewill write insymmetric form [1,2] where the indices off,.,start from n,0,subtracting 1from nandadding 1to0for thenext set,..,andfinally reaching O,n, Thus Ao=fooArsurfio+¥if,. Az=(U7/2)f2,0+ UiVifia +(Vj/2)fo2 +Uafy,o+ Vafo Ay=(U3/3Dfs,0+ (7/20 fay+(v7/2)uifa+(v1/3Dos+UUaf,o +(vith+UValla+ViValoa+Ushio+Valo, Acstaf o+Vafo +UT/2)uafs 9+(UT/2Dvofe +UrVitab, +uyvialig +(v7/2uafy 2HVT/2)vafos +UZ/2fao+ Uitste,o TVithbh y+Vata 1+UVof+ViVatoa +(V3/202 +(UBL ot(W/2D(UF/2Dfas+(UVSDE,a+UF/4DEeo +(V5/4 Dee ForAs,wehavec(0.0:0)= 1.WecanlisttheA,asfollows Ay=CU:ihe +0.1: Az=0(2,0:2)f 2+(11-206; 1+€(0,2:2)f 2+€(1,0;2)fyo+€(0,1:fo, Ay=€(1,0:3)f:,0+ €(0,1:3)fo,: +(2,033) otC(11:3)f,,5 +€(0.2;3)f,2 +€(3.0;3)f.0 +©(2,1:3)fp,. +€(1,2:3)f 2+0(0.3:3)f,s A= CL0:4)f, 0+0(0,1:4)fo 1+0(2,0:4)fa 9+CCL.4)f,1 +C(O.2:4)fe =+(3,024) o+(2,15a,+CULL:fy2 +CO.3ADE s+ 40:4) Eo+€B,1:4)f) +C(2.2:4)f2,2 +(13:4), 3+C(0,4:4)fo Ac=LOS): +CO1:S)fo,: +€2.0:5)f c+C(1,1:S)f,: +00.25) 2 +€(3,0:50f,9 +€(2,1:5)f, +6(1.2:5)f 2+(0.3:5)fo,s +064,0:5)f +3,19). +02.2:5)f22+ C135); s+00,45). +0(5.0:5)fe#C4,1:5)E.,.+63,2356 2+(23:5) +C(1.4:5)f.< +(0.5:5)fo.5 Ag=C10:6)f: 0+6(0,1:6)f,: +€(2,0:6)f: 0+C(1,1:6)f; :+(0.2:6)fo2 +.€3.0:6)f5,. +€(2,1:6)fe,1 +C(1.2;6)h 2+03:6), +o(4.0;6)f..0 +(3.1:6)f.) +(2,2:6)fy 2+C(1,3:6)h,» +C(0.4i6)fo,« +€(5,0:6)fsn+C14,1:6)fe1+13,256)2+C2.3:O)fe,3+C146),«+0.53605 +€(6,0:6)f.,, +€(5,1:6)fs,: +0(4,2:6)f,2 +(3.3:6)f5,s +O(2,4;6)fo,6 +(1S:60f, .+€(0,6:6)fo,6 ‘TueDecowrosmon MentooWwSeversDIMENSIONS 55 Ay=CLONE, 9+COLD o+CC20:7h,o +CULM, +00.2Dho 2 +(3,0:7) 0+2ATIfa,s+12: 2+(0,3:Mf,s+014,0:T)fi,o FABLDG +22M ha+CIM» +0.4:0f,.+ 5.0:Dfs,0 +(41M +B.2:)fi,2 +(2,3;ifa,s+CCA: +(0,5:7)fo,5 +(4.3: +CB.4:Dh a+C2,5:ifa,s +C(LE:TIE 2+0,7:T)fa,2 +€(6,0:T)fe,o +€(5,1:7)fs 1+664.2: +3.3:Nh 3+€2.4:7)fe,0 +ALS ME5+C0,6-ho,e +CT.0;TIh 0+C61; Mex +C5.2:7)I5,2 Ag=€(1,0:8)f, 9+(0,1:8),+€(2,0;8)f 0+€(1.1:8)f,,1 +€(0,2:8)f 2+€3,0;8)h 9+(21:8) 1+(1.2:8)f,,2 +(03:8) 5 +c(4,0:8)f,9 +€(3,1;8)f 1+€(2,2;8)f,2 +€(1,3:8)f),s +.0,4:8)f,« +€(5,0;8)fs,0 +0(4,1:8)f.,1 +6B,2:8)f 2+(23:8) +014:8)f. +0(0,5:8)fo,s +€(6.0:8)f5,0 +(5.1;8)fs,1 +€(4,2:8)f,,2 +6B.3:8)6 » +CAB) a+C(1.5:8)6 5+€(0,6;8)fo,5 +(7,0:8) 2+(6.1:8)f.,1+(5,2:8)fs2 +0(43:9)fi3 +CB.4:8)fs,. +(2,5:8)f,5 +11.68)F,,6 +€(0,7:8)fo,7+ €(8,0:8)f 0+0(7,1:8)f 1)+€(6.2:8)fs,: +0(5.3:8)f 5 +0(4,4;:8)fau +0(3,5;8)f5,5 +C(2,6;8)f,6 +0(1,7;8)f 7+.0(0,8:8)h 5 Ag=€(1,0:9)f,,9 +€0,1:9)fo,1 +€(2,0;9)f,0 +C(1,1:9)f,,) +(02:9) 9 +0(3,0:9)f5,0 +C(2,1:9)fa,1 +€(1,2;9)f, 2+€(0,3;9)fo,3 +0(4,0;9)fs,0 +CBLIE 1+(2,29)f,2 +(1.3:9)f; 9+€(0,4:9)fo,.+ 65,0:9)f,0 FOAL FOB2MG 2+€(23:9)f 2+.C(1.4: Fi.+000,5:9)f,5+(6,0:9)fo,0 +(5,1)fe,r+6(4,2:9)f.2 +€3,3:9)6 »+624:9)F FC(LS:9)fi,5 +0(0,6;:9)fo,6 +€(7,0;9)fr,0 +C(6,1:9)fs,1+ (5,2:9)F5 2 +C4,3:9)fa3 +349) «+25:90 5+C(1,6:9)f 6+(0,7:9)fo,2 +€(8,0:9)f5,0 +CTLMDE. +.(6,2:9)fe,2 +€(5,3:9)f,5 +064,4:9)fe,4 +3,5:9)f,s +26.9)fis+6170 7+0,8:9)f,s+ 69.0:9 +B LMF +720 2+C(6.3:9)fo 3+(5,4:9)Fs,.+664,5:9)fa,s +03,6:9)6 6+27» +(1,8:9F2+€(0,9:9)fo 9 Ayy=(1,0;10)f, 0+€(0,1:10)fo,; +(2.0:10)f 9+e(1,1;10)f, +€(0,2;10)fo,2+ €(3,0:10)f 0+€(2,1;10)f3,1 +€(1,2;10)f, 2 +6(0,3;10)fo,3 +€(4,0;10fio+6(3,1;10)f,.+ C(2.2:10)f,2 +O(1,3;100f, 5+€(0,4;10)f 1+€(5,0:10)f 0+C4,1:10)E.,, +.6(3,2:10)f 2+(2,3;10)fs,s +6(1,4:10)f, ,s+€(0,5;10)fo,s +€(6,0;10)f5,0 +€(5,1;10)f5,1 +(4,2;10)f2+€(3,3;10)6 5 56 curren 3 +0(2,4;10)f2 <4CCLS:10Vf5+€(5,2;10)fs,2 +4,3:10),5 +C(3.4;10)f5 1+C(2,5;10)s+C(1,6;10)f, 6+6(0,7;10)fo 7 +(8,0;10)fs,0+ C(7,1:10)f; .+€(6,2;10)f6,2+ C(5,3;10)fs,3 +€4,4:10)fa,4 +6B,5:10)b 5+C2610) +C(1,7310)6, 5 +€(0,8;10)fo,s +C(9,0;10)f o+(8,1;10)fs,1+€(7,2;10)f 2 +0(6,3;10)fe s+C(5,4;10)fs. +0(4,5:10)fs,5 +€(3,6;10)f5,6 +c(2,7;10)fs,7 +(1,8;10}f, 5+€(0,9510)fo,» +€(10,0;10)f0,9 +0(9.1;10}f,: +8.210) 2+C(7,3;10)f, 3+C(6,4:10)fe 0 +C(5.5:10)f5,s +€(4,6;10)f. 6+C(3,7;10)fs 7+C(2,8:10)F: +€(1.9;10)f 9+€(0,10;10}f, 10 ForA,=Dow c(H.v.1}f,,, weneedonlyc(1,0;1)=u, andc(0,1;1)= vy ForA,=32.(H,V.2)fyu Weneed (1,022)=us (0,12) =v; €2.0;2)=uj/2! (1,12) =u,v, (02:2)=vj/2! ASE, fe. €(1,0:3)=us €(0,1:3) =vs €2.0:3)=uju, (1,13)=uyv2+Uav, (02:3) =v,v2 6(3,0:3)=u}/3! €(2,1:3) =u}v,/2! o(1,2:3)=uyvj/2 (0,3;3) =v}/3! AeDicerCHVADE (1,0:4) =ws (0,1:4) =v. ¢(2,0:4) =u,vs+u3/2! ‘Tue DecoMPosimion MEmooINS EVERALDIMENSIONS 37 (11:4) 2uy¥5 +u3v +Usv; 0(0,2:4) =viv5+93/2 6(3,0;4) =ujuy/2! €(2,1:4)=uyusy,+uv,/2! (1,2:4) =uyViV2+ugv}/2! €(0,3:4)=v}v,/2! (4,0:4) =u}/4! ¢3,1;4) su}v,/3! ¢(2,2:4) =u7v7/212! o(1,3:4)=u,v3/3! (0,434)=v7/4tAsDives CYS)ie €(1,0:5) =us (0,1;5)=v5 €2,0;5) =uu U3, (11:5) =uyVe+gy+UV+UUM, €(0,2:5) =v,¥s +¥2y €(3,0;5) =03/2! +u?u,/2! (2,155)=uyuyv2+UyuyV,+?¥5/2!+udv/2t ©(1,2;5) =U,V,¥s +UavyVv,+UyV3/2! +u3v7/2! €(0,3;5)=vyv3/2!+v?v,/2! 0(4,0;5)=u}u,/3!€G,1;5) =ujv:/3! +uju,v,/2! €(2,2;5) =u?v,v2/2! +uyusv;/2! €(1,3;5) =u,v3/3!+uyv7V_/2! €(0,4;5) =v3vs/3! €(5,0;5)=uf/5! €(4,1;5) =ufv,/4! €3,2;5) =ujv;/2!3! €(2,3;5) =ujv}/2!3! (1,435)=u,vj/4! €(0,5;5) =v3/5! 58 Cuaron 3 A=TE, UMOla (1,0;6)=ug (0,1:6) =ve (2,0;6) =uyus+usu+U3/2! (1,156) =Uys +UaVe +UV +Wavy +USvs €(0,2;6) =ViV5+Vave+¥5/2! ¢(3,0:6) =ujuz,+u?u,/2! +03/3! (2,136) =uty+UysVs+UY,+UsUsV+ UT¥4/2!+UjV2/2! ©(1,2:6) =WV,Ve+WyV2Vy+UV,Vy+UgVyVg+ UV9/2!+WAP (0,326) =ViV2V5+VjVa/2!+3/3! ¢(4,0;6) =u?u3/2!2! +ufus/3! 6(3,1:6) =vjuju3/2!+vu;u;/2!+vguzuj/2! +v.u}/3! (2.236) =ujV3/2!2! +vPu2/2!2!4 uw?vyvy/2! +uyuyv/2! (13:6)=u-viv2/2!+wavy/2Mwavey3/2!+wav?/3! (0.426) =Vjvi/2!2! +vivy/3! ($,0:615 usu,/4! €(4.1:6) =uf,/4t +ufusw/3! €(3.2:6) =ujvi¥3/3!+ujuv;/2!2! C12.3:61= V}uju,/3! +v}vyu7/2!2! e(1.4:6) =viu/4! +v}¥.ui/3! €(0,8:6) =vy5/4! 6(6.0:6)=u5/6! (5.126) =ujv,/5! 0(4,2:6) =ufvj/2!4! €(3.3:6) =ujvi/33! 6(2.4:6) =ujv{/2!4! (15:6)=vu,/5! 6(0.6:6) =v{/6! A=Di Ye (10:7) =Ur €(0.1:7) =vy C(2.0:7) =Uy +UgUs +UU Tue DécouposmonMertoo1WSEVERALDIMENSIONS 50 CCDL;7) =UyVg#Us¥s+UY,+UAVs+UgVs+U4Y, (0,257) =VV +V2V5 +V5¥e (3,037) =uyuyuy +0,U3/2! +ujus/2! +upuy/2! 0(2,1;7) =vjuzuy +V2U,Uy +Vu;Us+V,U3/2!+vyu,Uy u}¥s/2! +vityus +vs/2! +vou;uy (1,237) =u,VzV. +UgVyVy+UAVV2+U,V3/2!+UsV,V5 $vFus/2! +UyVyVs +v23/2! +UaV2Vy (0,337) =V,VaV, +VV5/2!+VF¥5/2! +¥Z5/2! (4,0:7) =uju,/3! +uuu, /2!+uu}/3! €(3,1;7) =vyu}/3! +vaudu,/2! +vjujugu, +vzujus/2! +vsuruy/2! +v.uj/3! +vjuu;/2! ¢(2,2;7) =u}vyv,/2! +vjuyu,/2! +Ujvzv5/2!4 vfusu,/2! #UyUyVAVy +VyVous+Uyuav/2!+v,veuj/2! (1,337) =u,v3/3!+Uv}v,/2!+U,V,¥2V5 +ugy|vy/2!+Uyv7vy/2! +uv3/3!+U,vev7/2! 6(0,4;7) =v}v,/3! +VjVQV/2! +vv3/3E 6(5,0;7) =u}u3/2!3! +ufus/4t 6(4,1;7) =utu,y,/3! +ufujv,/2!2! +ujVv,/4l+ ufuy,/3! 6(3,2;7) =u}v3/213!+uyujv7/212!+usuavyva/2! +u}yvy/3! +ufuyv7/212! _ 0(2,3;7) =v}u3/213! +v,vju7/2!2! +v?veuu,/2! +v}u,u,/3! +vivyut/212! €(1,4:7) =v}vgu,/3! +v7v3u,/212! +viu,/4!+vvyu,/3! €(0,5;7) =v}v3/2!3! +vfvs/4t ¢(6,0;7) =ufu,/5! (5,137) =ufvs/5!+ ufuyy,/4! ¢(4,2;7) =uly,v,/4! +u}uvi/213! €(3,3:7) =ulv3v,/213! +v3utu,/213! ¢(2,4:7) =vjuyu2/4! +vfvu7/213! (15:7) =vfu,/5!+ vfvau,/4! €(0,6;7) =v} vy/S! ¢(7,0;7)= uj/7! ¢(6,1;7)= ufy/6! oo Charen3 e(5,2:7) =wiv2/2!5! 6(4,3;7) =ufvi3l4! c3,4;7) =viupBiat €(2,5:7) =uju;/2!5! €(1,6;7) =vfu,/6! (0,7;7)=vj/7! BeeChaves M8) (1,038)=Us €(0,1;8)=ve(2,0:8) =uyuy+Wade+usus+u3/2! (11:8)=uyVy+Uae+UsVs+sve+U5V9+UgV2+UGV) ©(0,2:8) =VyVp+¥2V6+V3V5+2/2! 0(3,0;8) =UyupUs+UUs,+UjU_/2!+UFuy/2!+upuy/2! (2,138)=vjugus+WyVet+UjU,Vs+VU, $+UyVety +UyUsY, +{UUs +76/2! +uzvou, +UZv,/2!+vytgu;+uFv9/2! (12:8)=uyVaVs+ViUaVs+¥;Vals+ULVSVe FVUAVe+V:Vu+ULV¥e+VjUg/2! $V2U:Ve +Vu./2! +usvav; +VFu,/2! (0,328) =ViVas +ViVav.+VF¥p/2! +vEve/2! +v2v2/2! 0(4,0;8) =uju/2!2! +uyusu,/2! +up/4l+ujusu,/2! +ufu/3! (31:8) =v,u,u2/2! +uFvyus/2! +vyuzus/2! +u,¥su3/2! +ujusuyv, +V2U3/3! +ujvjusus tuvou/2! ujuve/2! +vyuju/2!+uiv/3! 6(2,2:8) =uFv3/2!2! +viu3/212! +u,VviwsVs +uyuyv$/2! +v,vyu$/2! +uyuavavs +V;ValnUs up 3/212! +ujvavel2! +vjuswy/2! FUULV,Ve+VyVgU\UyU7VyV6/2!4 V7,Us/2! 0(1,3:8) =uy¥,¥3/2!+vjusvs/2! +wiv)v3/2!$V:UgV3/2! +vVyVallp+UQV3/3! +Vi)Va¥e $VTUsVal2! +vjVoug/2! +U:ViVs/2! +VpUs/3! TueDecouPosimiow MMODINSEVERALDIMENSIONS 61 (0,438) =vv3/212!+vivivs/2! +v3/4! +vivav/2! +V¥5/3! 6(5,0;8) =u7u3/2!3! +u}u;u,/3!+upu,/4! (4,1;8)=v,uju 3/3!+u}udy,/2!2! +vu?uus/2! +UfV;uy/3! +ujuyvs/3! +ufyu/3!+ uty,/4! 6(3,2:8) =u}vj/213!+ ujuv 3/212! +vyv.u,u3/2! +{uyuUs/2! +u}¥,vu,/2! +UTVitvs/2! +ujvavs/3! +Uvve/3! +upuy;/2!2! 0(2,3;8) =v3uj/213! +v}vzud/2!2! +uyuzv,v 3/2! +U2v,¥v5/2! +vPuyuyvs/2! +v7u,vzus/2! +v}uyus/3! +v}uju,/3!+ viv,uz/212! (14:8) =uyv,v3/3!+viV3u,/2!2! +uyvjvv,/2! $V[Ugv5/3! +Vv}vgus/3! +v{uyVa/3H+vful! 6(0,5;8) =v}v3/28!+ Vjvave/3! +viva/4t €(6,0;8) =uf'u3/2!4! +ufu/S! ¢(5,1;8) =v,uju3/2!3! +vu;u,/4! +v,usuy/4!+ ufy,/5! ¢(4,2;8) =ufv3/2!4! +vjuju3/21212! +V,VqU}W2/3! +ufyvs/4! +uv;Us/213! €(3,3;8) =u}y,v3/213!+v?u,u3/2!3! +v?v,u,u?/2!2! +uiviv,/2i3!+ vu?u,/23! (2,438) =vfuj/2!4! +ujv}v3/212!2! +Ut] V2/3! +vfuu/4!+VUv4/213! (1,538) =uv}¥3/213!+Uv}vy/4!+wiv;vy/4!+ Vvfuy/5! €(0,6;8) =viv3/2!4! +viv,/5! 6(7,0;8) =utu,/6! ¢(6,1;8) =ufv2/6!+v,uju,/S! €(5,2;8) =u}v,v,/5!+ vjuu{/2!4! 6(4,3;8) =usvivy/2i4! +uv}uy/3!3! €(3,4;8) =vfuju,/2!4! +v}u}va/313! €(2,5;8) =vju,u/5! +ujv,v4/2!4! C(1,6;8) =viU,/6! +u,viv;/5! €(0,7;8) =vfva/6! (8,0;8)=u¥/8! 2 Cuavron 3 (7,1:8) =ujv,/7! (6,2;8) =uSv3/2!6! (5,358) =u}v3/315! (4,438) =ufv;/4t4t 6(3,5;8) =vjuj/3!5! 6(2,6;8) =vfu;/216! o(1,7;8) =v[u,/7! (0,8:8)=4/8!ASDerCMDNE (1,09) =u, (0,1:9)=vy €(2,0:9) =uyUy+usu,+st+Usd, C(1.1:9)=Uy¥y+UsVs+UsVs+UAVs+USNs+UEYy+UsVy+UF 02:9)=vy;+VEYEVE,+Ns 6(3.0:9) =u.u7/2!+ujus/2!+uuu+3/3!+u7u,/2! +U,usu, +uuu, (21:9)=v,UH/2!+Uytavg+ve/2!+UVotls+Vet,+Usyth +UUsVg+V3U5/2! +V,UyUy +UfV9/2! +¥,UUs FUVaUg+UULVe+Vly+ULVls+0)U,¥5 0(1.2:9) =uv3/2!+v,vau,+V2ug/2!+VataVs+UZVyVe+ValVe VGNgle +UV3/2!+UyypVy+V2u5/2! +ULVEVe FV UV +, Val, +UVAs +10,¥5+VNUs 0(0,3:9) =vv/2!+v3vs/2! +vavgVe +V3/3!+v2vy/2! +V,ViNe FVIVBVs 0(4.0:9) =u3us/3! +uyu,u3/2!+u,-uju,/2!+u7uu,/2! tufujus/2! +u}u/3t 6G.1:9) =u}v,/3! +vjuzu,/2! +v,uju 3/2!+u,vu3/2! +uu; +yyudus/2! +ujusvsue +upudve/2! +vst +uivyus2! +uPuyv4/2! +vjujuaus +UP¥gUs/2! +U[UnVs/2! +ulve/3! +vueu/2t €(2,2:9) =wivyv5/2! +v3uyuy/2! +v,vqu 3/2!+uyusv 3/2! FV,VatUs+UjUVZV, +¥,v,U3/2! +uyuve/2! +UUVave+VyVauuy tu?%yVv,/2! +v7uu/2t $+Vyvyujus +yuyive+Pups!2!+U2viv5/2! UUV;Vs+V;VayUsFUPv,v6/2!+vPu,U6/2! 6(1,3;9) =v}uy/3! +u,v3v,/2!+uyvav5/2!+vityV5/2! $V,VpVyHuyViv4/21+V, VatVe+VV5UL/2! FULVyVaVe#V;UsV4/2!+VjVyte/2! +UyViVes FV[UyVs/2! +VPvzUs/2! +vfug/3! +U,V;V6/2! c(0,4:9) =v3v5/3! +v,v.V3/2!+vyV3V4/2! +VjVyV4/2! FVivavs/2!+ viv/3! 6(5,0;9) =u,u3/4!+ufulus/2!2! +ujus/2!3! +ujuu/3! +ufus/4! 6(4,1;9) =v,u3/4! uyvyu3/3!+V¥,u,u5us/2! +Ujuyv-us/2! +ulujy,/212! +v,ujuj/2!2!+ulvus/3t +vyufusu/2!+uivju/3!+ ulusv/3! $v,wfus/3!+ufys/4! 6(3,2;9) =uv;vg/3!+uyw3v3/22! +vju;Uy/2!2! tu}yiuy/2!2! +v,vyu, 3/2! +v,v2u)UyUy +vavyu?u/2! +u?v5/213!+v2u, u3/2!2! +V,Vgu7uy/2! +v7u,ugu/2! +v2v07/3! +vyvququl/2! +v,vag uy/2!+vFuzu/212M Vv,vsui/3! (2,339) =v}uyu,/3! +v,vju3/2!2! +ufvgv5/2!2! tvPudvy/2!2! +uyuyy,v3/2!+wyUyvyvEV3 +ugWyviv,/2! +v}uj/213! +uFy,vj/212! +UyV7Vy/2!+UyvaVe/2! +UgUV/3! HuyUyVav7/2!+UyUvvg/2!+uPv;v5/2!2!+ uyUsv3/3! (1,459) =u,v3/4! +v,Upv3/3! +vyV3V9/2!+VfvaVs/2! +viviuy/212! +u,vivi/212! +vtu,vs/3! +uyVivavy/2! +vfusvi/3! +v}v2u4/3! +,vivs/3! +vfus/4t 6(0,5;9) =v,vi/4! +viviv9/2!2! +v}v3/213! +Vivavi/3!+V fvs/4t 6(6,0;9) =u}u3/3!3! +ufuu,/4! +ufu/5! (5,1;9) =v,u}u3/213! +v.u}u3/2!3! +ufv,uy/4! +ufu,y)/4!+ Vv,u}usus/3! +vyui/Si+v,usu/4! “ Cuaron 3 6(4,2;9) =viuu3/2!3! +v3uyu}/2!3! +v,veu;u3/212! +vavyus/4! +vjuju;u/2!2! +v,vyutus/3! VyVguyur/3!+v/u;u/213! +v,vas/4! 0(3,3;9) =u}v3/3!3! +v{u3/3!3! +ujuyvzVv;/212!+uiuyV;va/2!2! +}V,¥2¥3/3! +vtu,usuy/3! +U2viuzvs/2!2! +ufvivous/2!2! +ujy;ve/213!+vpu;wy/2!3! 0(2,4:9) =u?v,v3/2!3! +utv,v}/2!3!+u,u,vivi/22! +upuy{/4! +u}v3v,vy/2!2! +uyusviv3/3! uyWv{V2/3! +ujv;v./2!3! +uymvs/4t (15:9)=wyv3v3/2!3! +u,v}v3/2!3! +vfusvs/4l $Vf¥:Us/4! +Uy;Va¥5/3!+vy/5!4 U,VivA/4! 6(0,6:9) =v}u2/3!3! +v{vavs/4!+viv/5! ¢(7,0;9) =uu/2!5!+u fus/6 €(6,1:9) =ulusv2/5!+u suzy/2!4!+fy3/6!+uSusy,/5! 6(5,2,9) =ufvi/2I5! +ujuly;/2!2!3!+ufu,vy,v2/4! sulvvs/St+usuy3/2i6! (4.3.9) =uty, vi/2I! +uluzvi/228! +ufu.v?v,/23! +ufvivs/214! +ujuy}/3!3! 6(3,4:9) =fu,uz/214! +v?v5uj/21213! +vfvguzu,/2!3! +vfupus/2!4! +vivsuj/3!3! 6(2.5:9) =vivi/2!5! +vjviuZ/2!213! +viveu;u/4! +viuyu,/5!+ vivyuj/2!4! €(1,69) =vivzu,/5! +vfv2u,/214! +v6us/6! +vivsu/5! €(0,7:9) =vivi/2!5!+v fv3/6! (8,09) =u/u./7! (71:9) =uly./7!+ ufuy/6 €(6.2:9) =u‘ v,¥,/6!+Survj/2!5! (5,339) =uivi vy/215! +ufu,vj/3tat ¢(4,4:9) =uly v{/3!4! +v}vjuf/3!4t €(3,5,9) =viuiu,/2!5! +vjvpuj/3!4t (2,69) =vSu,u,/6! +vfv,u7/2!5! (1,7;9) =v7u./7! +vfv2u,/6! Te DécouposiioN MErt00iSEVERALDIMENSIONS “6 (0,89) =v/v4/7! (9,0:9) =u?/9! €(8,1;9) =uyy,/8! (72:9) =ujv7/2!7! €(6,3:9) =u’v}/3!6! (5,459) =uivi/415! (45:9) =ujvi/ais! 6(3,6:9) =u}v{/3!6! €(2,7;9) =ujv,/2!7! (1,8:9) =u;v}/8! €(0,9:9) =vj/9! CONVENIENT RULES FOR USE: The A, have been written indetail asaconvenient reference andanaidin calculations. However, they cannow bewritten bysimply remembering the algorithm. The c(jt,Vsn) arewritten byconsidering allpossibilities for and vwith +Vv=n.Inspection ofthe listed c(,vzn) will make itclear that tells ushow many times uappears and vtells ushow many times v appears. Further, weseethat thesum ofallthesubscripts ismand aswith functions ofasingle variable, repeated indices require division bythe factorial ofthenumber ofrepetitions. ANALYTIC FUNCTION OFSEVERAL VARIABLES: Let's consider f(u,v,w)#f(u)E,(v)f,(w). Thus,N[u.v.w), withNanon- linear operator, acting onuisananalytic function f(u,v,w) which weset equalto1,A,.Nowwedefine faves=(2*/Ato)(3/A¥0)(3*/305)f (os¥oro) Now Ao=fo.n0 A,=(1.0.0: fiao+€00,1,0:6,00,0,151)F5 Az=C(1,0,0;2)fi 00+€(0,1,0;2)f9,1,0+ €(0,0,152)fo,0,1 +0(2,0,0;2)f2,0,0 +€(11,0:2)f, x0+(10,1220 on +€(0,1,1;2)fo,41 +€(0,2,0:2)fo 20+€(0,0,2:2) 02 66 Curren3 Thevaluesofthec(J1,V,@) aboveare (1,0,0;1) =uy €(1,1,0;2)=uv, €(0,1,0;1)=v,€(1,0,1;2)=u,w, €(0,0,1;1) =w, (0,1,1:2)=yyw, (1,0,0;2) =u, (2,0,0;2) =u7/2! €(0,1,0;2) =vz (0,2,0:2)=v3/2! c(0,0,1;2) =w2 £(0,0,2;2)=w7/2! Thus Ao=flUoYo,Wo) A,=u,(9f/du,) +v,(8£/2 vo)+w,(9£/22,) A,=u,(0£/duy) +v,(A£/d vo)+w3(Af/92,) +u,y,(3?£/du,dv,) +uw,(d?£/9u,dwe) +yyw,(2?£/9 v9wo)+(uj/2)(9£/2u3) +(vi/21)(a £/av5)+(w;/2(2 f/aws) etc. for A. We can proceed analogously fordetermination ofA, for functions f(uy,uz....s)- APPLICATIONS: The A,forfiu,v,w) isneeded tosolve three coupled nonlinear differential equations. Inthe author's form [2} for coupled equations, using decomposition wehave usLy'g)- Li'R,(uy.w) -LN,(uw) v=Ljg:—L]R,(uv.w) —L}N2(u,v,w) w=Lj'gs— L3Ri(uy.w) —L3'Ns(u,v.w) Weletu= D7,uy,veEZ,vy.w=EZ,w,andwewriteN(uv.w)= fuvw)= D7, A,{f(u.v,w)} fori=1.2,3.Then Uy=0,+Li'g,where L,o,=0 Vo=®,+ Lig: where L:0,= 0 Wo =;+Lj'g,where L,o,=0 Similarly werequire A,{flu,,...tq)) formcoupled operator equations Anexample foranon-factorable nonlinearity f(u,v) isthepair ofcoupled TeDecouposmon MerionwSevekatDIMENSIONS “0 equations du/éx +au +bv +flu) =, dv/dx +au +bv +f,(uy)=g, Finally, weconsider N(u,v) =f(u,v) =e'**. This isaninteresting case for comparison purposes since itisafactorable nonlinearity: e*"=e*-e",sowe cansolve itasaproduct nonlinearity using A,(f(u)} orwith thepresent results for A,/f(u,v)}. Wecannow consider asetoftwo coupled equations inthegeneral form: Lu+R,(uv) +Nw)=8) Liv+R, (u,v) +N(u,v) =g wheré N(u.v) =e SOME FINAL REMARKS: The definition oftheLoperator avoids difficult integrations involving Green's functions. The useofafinite approximation inseries form forthe excitation term, and calculation only tonecessary accuracy simplifies integrations still further. (With Maclaurin expansion, forexample, of trigonometric terms, oneneeds only integrals oft°.)The avoidance ofthe necessity forperturbation andlinearization means physically more correct solutions inmany cases. Theavoidance ofdiscretized orgridmethods avoids thecomputationally intensive procedures inherent insuch methods. The decomposition method iscontinuous and requires significantly less processing time forthecomputation ofresults, Ithasbeen demonstrated that very few terms ofthedecomposition series arenecessary foranaccurate solution, andalso thattheintegrations canbemade simple bythesuggested methods, orbysymbolic methods, andusequite simple computer codes in ‘comparison with methods such asfinite differences orfinite elements. Aswehave shown, partial differential equations canbesolved bychoosing oneoperator fortheinversion andconsidering allother derivatives tobein- cluded intheRoperator. Hence wesolve exactly aswith anordinary differ- ential equation. Wehave theadditional advantage ofasingle global method (forordinary orpartial differential equations aswell asmany other types of equations). The convergence isalways sufficiently rapid tobevaluable for numerical work. The initial term must bebounded (areasonable assumption foraphysical system) andLmustbethehighest-ordered differential. 68 Carre 3 REFERENCES 1. G.Adomian, Stochastic Systems, Academic Press (1983).2G.Adomian. NonlinearStochastic OperatorEquations, Academic Press(1986).3. G.Adomian andR.Rach, Purely Nonlinear Equations, Comput. Math. Applic., 20, (1-3) (1990). 4. G.Adomian andR.Rach, Equality ofPartial Solutions intheDecomposition Method forLinear orNonlinear Partial Differential Equations, Comp. Math. Applic, 19, (9-12) (1990) 5. G.Adomian and R.Rach, Noise Terms inDecomposition Solution Series, Comput ‘Math.Applic..23,(19-83)(1992). 6. G.Adomian, Solving Frontier Problems Modeled byNonlinear Partial Differential Equations. Comput. Math, Applic.,22, (91-94) (1991). 7. G.Adomian. R.Rach, andM.Elrod, OntheSolution ofPartial Differential Equations swith Specified Boundary Conditions, J.Math. Anal. and Applic., 140, (569-581) (2989) 8. G.Adomian and R,Rach, Generalization ofAdomian Polynomials toFunctions of Several Variables. Comput. Math. Applic, 24,(11-24) (1992). SUGGESTED READING 1. N.S. Kosblyakov, M.M.Smimov, and E.B. Gliner, Differential Equations of ‘Mathematica! Physics, North Holland (1964). 2. M.M. Smirnov, Second-order Partial Differential Equations. S.Comet (ed.). Noordboof(1964). 3.EA.Kraut,Fundamentals ofMathematical Physics,McGraw(1967). 4N.Bellome. 2.Braezniak, LM.deSocio,Nonlinear Stochastic FvalutionProblemson Applied Sciences, Kluwer (1992), 5.A.Blaquitre. Nonlinear SystemAnalyses,AcademicPress(1966). CHAPTER 4 DOUBLE DECOMPOSITION Insolving boundary-value problems bythedecomposition method, wehave seen thatwecaneither retain the“constants” ofintegration intheuyterm for the caseoflinearordinarydifferential equations, re-evaluating theconstants as ‘more terms oftheapproximate solution g,arecomputed, or,wecanusethe tuevaluated tosatisfy theboundary conditions andaddconstants ofintegration foreach successive term u,. Foralinear ordinary differential equation, itismore efficient tocalculate an n-term approximation @,,carryingalongtheconstants us,andfinallyforce@, tosatisfy theboundary conditions, thusevaluating theconstants ofintegration. Wenow introduce aneffective procedure which allows usdecreased computation, especially inpartial differential equations. This isdone bya further decomposition, i.e.,wenow decompose theinitial term u,=®into Ti, Parieste=D2, Yow[I Atfirst thought, thisseems likeanill-advised procedure which canonly slowconvergence, sincethenewinitialterm®,OrUgowillbefartherfrom theoptimum value forus.However, wewill seethat, asaresult, wecan use ©,todetermine a which canthen beused forfurther terms ofg,without further evaluations. The boundary-value problem becomes anequivalent initial-value formulation interms of©.This eliminates further matching to boundary conditions. Letusagain consider theequation u,,-u,,=0 with u(y) =0, u(x,0) = 0,u(x2,y) =siny,andu(x,1 /2)=sinxwhose solution bydecomposition is u(x,y) =sinxsiny.Wewillagain usedecomposition andalsotheconcept of equality ofthepartial solutions oftheoperator equations, soonly oneoperator ‘equation needs tobeconsidered. Also, wewill decompose theuyterm ofthe decomposition series, which means adouble decomposition ofthesolution u. (This isamuch preferable method tothatofeigenvalue expansion inm dimensions.) WehaveL,u=Luandcanapply theinverse operator L;’onboth sides. ‘Thus L7'Lu=u-, oru=,+ Lj)Lyuwithu(0,y)=0andu(/2.,y) =sin y.Equivalently, wecanstartwithLu=Luandapply Lj’towrite u=&,+ L;'Lyu with u(x,0) =0andu(x, 2/2) =sinx. 69 70 Cuapren 4 Asusual, weassume u=.”_, u,butnowwealsodecompose usinto Dz, Yon:Forthexconditions, wehave Yvan dLwet, we nm = withup=,» andu,,,=. +L;'L,u,.,. Wecanalsowrite, using they conditions, theequation Dred Met GLY ve mm Ea] withup=,, anduz,=0,,, +L;'Lu,.y. Since L,, =0andL,®, =0, we have 0 =Eo(y)+x6(y) =Syn) +6,09) ®,.9 =M(x)+YT,(%) = Maa)+Mw(X) where the£'sand 7'sarise from theindefinite integrations. Theconditions given determine these integration “constants” fortheapproximate solution Paes=ng YerThus y.,(Oy)=Oand 9,,,(8/2,y)= sinydetermine Suu(y)and&,.(y). Similarly, ,.,(x,0)=0 and9,.,(x,2/2)=sin xde- termine 7,(x)and7),,,(x). Letusconsider improving approximations tothex-dimensional solution as wecalculate increasing terms ofthedecomposition series. Ofcourse, the approximation isthesolution inthelimit m—>o us, +LLu=g+xG +LLye Y=Uy=Soo XSi Since 9,(0,y)=0, &.=0. Since 9,(4/2,y)=sin y,$,.=(2/x)sin y. Therefore, 0,=uy=(2/z)xsiny. Tocalculate u,wehave ovate Decourosirion 71 =Gy+XE,4 +LLuy Lu, =-(2/)xsin y LitLyu,=-(2/2)(x°/3!)siny uy=,+xG,q~ (2/2) /3!)sin y ‘Avwo-term approximation isgiven by9,=@,+u; (oru,+u,); hence 02=(QUm)xsiny—(2/a\x /31)siny+S,=x3, Since 9.(0,y)=0, wehave ,,=0, andsince g,(/2,y)=ssin y,wehave Sug=(/2\(sin y)/3! uy=(/2)(x siny)/3!'—-2/ x)(x°siny)/3! uy=S52+6, +LEL,uy L,u,=(2/m)(x? siny)/3!-(e/2)(x siny)/3! LiLu,=(2/2)(x° siny)/5!-(1/2)(x° siny)/(G3!)* uy=Ena+xG,2+(2/2)(x° siny)/5!-(2/ 2)(x°’siny)/G!°9,=0,+0;OFUy+,+U, etc,Summarizing, thecomponents ofuare u,=(2/7)xsin y u,=(#/2)(x siny)/3!-(2/2)(x’ siny)/3! uy={-(@/29 /51+(2/29 GB!)}xsiny —(#/2)(X siny)GIF +(2/2)(X* siny)/5! etc.Theapproximate solutions 9,,9:,y,... are: Q,=(2/x)xsiny x=(2/m+(0/2)/3!)x siny+(2/#)(-x?/3!)siny y=(12)+(02/2)/31-(8/ 2)51+(2/2)(G1)*)x siny +112) +(@/2)(3!)(-x? /3!)siny +1/(r/2)(x5/5!) siny 2 Churren 4 etc, oF ,=(.6366198)x siny 9,=(.8984192)x siny+(.6366198)(—x° /3!)siny 9,=(9737817)x siny+(.8984192)(—x? /3!)siny +(.6366198)(x°/5!) siny which converges very rapidly tothegiven solution. Itisinteresting towrite the resultas Pq=AnoXSin¥+8_(-X°/3!)sin y+a,.(x°/5!)sin y+ or Pn=>Baal") 20+)! sinyFd where thea,,, arenumerical series whose sum is1;each term isdelayed behind thepreceding term. Now, a san| limg,=lim¥a,.,(-1)*08*" fen+1!siny pone~gin2 wherelima,.,=1foralln.Then u=lim@,=D{Cp°oe*)/2n +}siny=sinx-siny The y-dimensional solution isu=sinysinxsince, bysymmeuy, yis imterchanged with x;ic.,thepartial solutions areidentical. Consider theexample Uy+Uy=g(x,y) =x*+y? with u(0,y) =0,u(x,0) =0, u(Ly) =y*/2, u(x,1) =x°/2.Wehave shown previously, using decomposition, thatthesolution u=x’y’/2 canbeobtained inonly twoterms. Itisalsoclear thateither theoperator equation forLuorforL,ucanbeused with appropriate inversions. Thus Lu=x'+y?-Lyu LyLu=L* +y?)- LeLu andsince L'L,u=u-, Dovsus Decourosirion 7 usO,4+Li(x+y?)-LiLu 7) Similarly, uso, +Li(e+y*)-L Lu @ Using (1), uy=O, +L3(x° +y") DY4-4-LiL, Yu Ss = uy.=-LyLyu, form >0. Now, ifwedecompose theupterm aswell, wewrite Y= LO. +y)-LIL, Du Identifying u,=©,,+L;'(x’ +y*),allothercomponents aredetermined by Un)=O.) -LiLyuy cd) Proceeding analogously using(2) Uy=O, +L(x?+y*) : a Uae=Pas LyLvs Continuing with thexequation, i.e,(1)and(3),” ©,=Gly)+xG(y) “6i=SaalYWXE,a(¥) ‘from (2)and (4) 0,=>=N(x)+y7,(x) oy yy=Masa(X)+YT(*) The“constants” of(indefinite) integration arenowmatched withthe approximate solutions g,forn=1,2...wherePuui=SuevgUsThus Pa.(0y)=0, Pau(ly)=y'/2 determines &,(y)andé,,(y) in(5). Similarly, 9,,,(x,0) =0and$,4:(X,1) =x°/2determines 7,,(X) and1),(X)- ” Coarren 4 Proceeding with thex-dimensional solution, ®,=&,(y)+xé(y) and Uy=&+x6,+L3'(x? +y*);afterdecomposition ofup, Ue=Foot xSio tL(x?+y*) Uaer=Somme XSrae)“LyLye Ourfirst approximation is@,=u,,or =Soot XSi9 FXM2+x7y7/2 where 9,(0,y)=0, @,y)=y*/2. Since 9,=(Oy)=0, &9=0. Since O(Ly)=72, Eyt/12+y?/2=y°/2 or,.=1/12.Hence uy=—x/12+x4/12+x°y7/2 Then u,=55,428, -LiL,u, SinceL,u.=x*andL;'L,u, =Ly!x*=x4/12 w=Sy) XSMID Then = Uy+t, =9,+U, ‘ayn 4 afZEA lexz2X -|1 2prise*u-Dl eeneeT Rabeta Since 0.(0,y)=0 £20 o,(Ly) =y?/2 5.51/12 uy=x/12=x8/12 Dovsue Decournsinow 7s Wenow have g,=xy?/2, i.e.,theexact solution inwoterms. Ifwe proceed further w=G24x6. -LyLu, Wehave L,u, =0,Li'Lu, =0 B=, tu,=YDS 2$HS2 andsince 9,(0,y)=0, 3=0.Since ,(I,y)=y"/2, §,.=0; hence u,=0 s09,=x'y"/2, Wecancontinue toseelimQ,,,=u=x°y*/2, Thesame result isobtained from they-dimensional solution. We now apply thedouble decomposition toalinear ordinary differential equation represented byLu+Ru=gwhere Listhehighest-ordered linear differential operator—in thisexample wechoose L=d’/dx’andRisalinear operator (the“remainder” operator) which cancontain forthisLnoderivatives higher than thefirst (the order ofRisalways lessthan theorder ofL).. DIRICHLET CONDITIONS: u(b,) =f,and utb;) =p, Solving forLuandoperating withL",wehave u=®+L"'g-L™Ru where L®=0.Nowlet u=0,u,and=", ©,;then Yuw=Y +L 'g-LRYvy, a ot = (where Lis apure integration not involving constants). Let ©,=Cog+X,_ and define u,=,+L"g. Now ©,=) +XC5. Matching ,totheboundary conditions c,and¢,.aredetermined bytwo linear equations. Suppose g=0forsimplicity. Then C09*PiC,9 =By C0+BCo =B: orinmatrixform 76 Cnarrex4 1bi]feoo]_[B, 1b.) Lew) LB, ifthedeterminant ofthefirst matrix isnon-zero. We now gotothenext approximation g.byfirstdetermining u,=, -L"Ru, toget@,=y,+u,. Matching 9,totheboundary conditions toevaluate the constants, g.isdetermined completely. Continuing inthis manner, we determine u.,u,,... until wearrive atasatisfactory 9,verifiable by substitution orstabilized numerically tosufficient accuracy. Wehave u,=®, -L"Ru,_, where ®,=c;,,.~XC,,, andQ,,.,=@, +U,. Matching @,,, totheboundary conditions, werequire ari(0)= a0)+Ug(0,) u,(b,)=,(b,)-L"Ru,_,(b,) where ®,(b,)=C;,. +b,C;,q. Substituting andmatching theconditions, Cog +DiCg—L"Rug(04)+Pa(b,)=B, c,h, -L'Ru,_.(b-)+9,(6,)=8, Rearranging. ComFDC =B,-Pq(B,)+L"Ruy .(0,) =Bye Cin =0:61 =By~Pq(bs)+L"Ru,(0)*Brn which wewrite simply as Com+PCie=Bie Com +DC,.0=Baw or [}b,]fee]_[Bie]Lblle |Loss, where Dovste Decourosrrioy 7 [Bim] _[B~Pa(b))+L"Ru.(b,)] Boa!”|B:-oa(b;)+L"Ru,,,(b.) | Thus, fom]_f?hal Lea}[2be(a| Now ®,ofC,,,and ¢,,,aredetermined andweremark that con] 1lim[oom] limBaloo emloat 8 [Bre ‘Thedecomposition oftheinitialtermcanbeusedfornonlinear boundary-value problems (for ordinary orpartial differential equations) andalso forlinear partial differential equations. Itisnotnecessary inlinear ordinary differential ‘equations where wecancarry along theunevaluated upandevaluate allatonce intheQ,asimpler procedure. Theobjective ofthedecomposition ofuyisto allow aconvenient matching oftheboundary conditions toanyapproximant Gaie.,foranyvalue ofm.Each integration involves constants which are added togetabetter uy.This gives usauseful procedure. EXAMPLE: u,,+1, =0withtheconditions ua, y= aly) - uaz-y)=@(y) u(x,b,)=B(x) U(%,,)=B(x) WriteL,u+L,u=0. IfwesolveforLu,wehaveu=®,-L7!L,u where ©,=&(y) +x6(y). Nowdecompose ®,also;thus©,=)”, ®,,,.Then Up=So+XE,0 uy=Soy+S,“LAL,(Goo+X5,0) us=Gyx8,LIL,Gy+xG)+LYL,)(Gyo+6,0) Ug=D (LIL, Gare+Xmas) Fd 7 Coarren 4 where €,,,and&,,,aredetermined bysatisfying theboundary conditions with theapproximate solution @,,,=D), u;;thus, Gaei(@¥)=@,(y) Gaei(@zry)=On(y) ‘Thesolution is), u,,or =DD LY Cane +See) ao ot w=DD CN/201)(9 149"nwo) 2X DCM/20+D)(9"/49™ Ewen) at ob v= -1"(/om)(a*/a*)E, 4.0) =Yeens Yaa) &.09 Since wedecomposed u,, SM= >£0) ém= >&.0) sothat weobtain the solution w=DCD(e*/(2nynfa*/ay** Soy) Laan =ya/a*Ey) Wehave seen that thesolution canalso beobtained from theequation forL,u. Thus, ifwewrite Liu=-L,u and apply theinverse L;!wehave us, -L/L,u where Dove Decourosimion 79 ,=M(x)+T(x) Now u=n,(x)+yn,(x)-LyL,u where L,=0?/dx andL;'is atwo-fold indefinite integration withrespect toy.Weletu=." u,where upis normally given byTo(x) +yn,(x). Wenow decompose theu,also, i.e., Y= Meo +¥ho Y=MesFM LL[Mo+M0] Us=Meat¥Mha “LLMs +YM] +(L7L,)[Moo+vt] Ug=L(-LFL.) [Mowe+¥Tha-s] where Gaoi(%-b,) =B,(x) Gavi(Xsb2) =B,(X) Nowu=lim@,,,.Intheinhomogeneous case8#0, uy=O, +L'g - u,=®,-(L"R)L"g-(L"R)®, ug=,(-L"R)'®,_,+(-L'R)"L"'g Ea} Finally, summing togetthesolution v=3S (ewye.+(Ry L's Rearranging terms, s0 Charron4 us (-L'R)"Y o,+(-L'R)*L'g a a u=>(L'R)"{o+L"g} = The value ofdecomposition oftheinitial term isthematching ofthe boundary conditions foreach @,forany m,Every integration has new constants toevaluate. Finally, wecanaddalltheCoqandCc),Separately to form anew cyandc,orequivalently, anew uywhich now isclose toafinal value which would bereached as ne in¢,. NONLINEAR CASE: Consider theordinary differential equation Lu+Ru+Nu=g.Solving for Luandapplying L”: u=c,(y)+xe,(y)+L7g-L"Ru-LNu (Again L’'isanindefinite integration—in this case, two-fold.) Since ©,= co(y) +xC\(y) isdecomposed, Up=Oot L'g u,=,,-L"Ru, +L°A, Up=O,_~L'Rug.) LAs; Thesolution u=ju, and®,=7,®,., where Do =Com¥)+XC.n(¥) NONLINEAR CASE—PARTIAL DIFFERENTIAL EQUATION: Consider Lu+Lu+Ru+Nu= g.Weassume that L,=2°/a* and considertheequation forLiuwithL,treatedasanotherRterm.Iftherearealsooperators L,andL,,theyaretreatedexactly likeL,;then, Dovate Decourositon a L,u=g-Lju-Ru-Nu u=, +Lg-L{Lju-L{Ru-LNu where ®,=c,(y)+ xc,(y). Decomposing ,where ,=c,(y)+xc,(y), Uy=epg RCo HL u,=Cy)xe),+L'Lyu, -L3Ru, -LA, Toevaluate, wehave ,=u, which ismatched totheboundary conditions. Suppose thexconditions areu(b,,y) =8,andu(b,,y) =8.Then Sr0+DC,9+LB=B, Coo+Baty +LB=B sothat Coo+DCy0 =,-Lig Con+batyo =By-Li'g or 1by] [e0]_[ 8,-Lig 1br} Go) [6-Li'g, from which wedetermine c,,andc,o,sothat9,=uyisdetermined com- pletely. Now u,iscalculated from uy=Cy,+x¢,,-L7Lu, -LYRuy -LA, ‘Since uphas been determined, u,iscalculable, soweobtain $,=9,+u,which ismatched totheboundary conditions using Coy+bie,~LLUy—LERuy—L3'Ag+6,(,)= Go+Bae, =LfLyLRU Lg +6,(b2)=By sothat4,isknown, Theprocess iscontinued toasatisfactory ¢,. 2 Cuarren 4 Consider the equation Lu+Ru=0 where L=d%dt* and Risalinear operator possibly involving differentials oflower order. The integral representation is: u=®-L'Ru where L~'isanintegral operator defined asann-fold integration and u=>2, u,yieldsthesolution inseriesform.Itisinteresting toconsider adoubleseriesrepresentation u=% Dtes ee2Yes eae IfLisofnthorderinthesingleseries,wehaveu,=®whereL®=0and® hasnterms. Suppose thenwedecompose =)”, andletu,=, only, Now uj,which waspreviously identified asu,=-L"R@. becomes u,=, -L'RO, u;=©,-L"RO, +(L"R)', uz=O,-L'RO,,,~...+(-L"R)""®, +(-L'R}°o, Now ¥v=>o,-LRY ©,+...+(-LR)o, ‘Theapproximate (m-term) solution isgiven by: On.=Y,O,-LRY, ©,+...4+(-L°R)” Yo,-(-L"R}"o, whichwecanwriteasadouble summation Dovare Decoupostrun a aact on MY Y(ERO, at i aS =3(-'R)"S 0,=S(-rjro- u soweseethat ourm-term approximation becomes theexact solution inthe limitasexpected. Wecannow view initial-value andboundary-value problems inthesame way, offering clear advantages over finite difference orshooting methods. Thus ininitial-value problems, u,=-L'RO u,,=-L"R®, byUs=D o,a B weClRyo u,=(-br)o, YSu,,=(41'R) yo, Ed En u,=(-L'R)"® u,,=(-L'R)"o, >Uns=(RP >o, The approximation ¢,isgiven by el ot a=D(-E'R)DEa} a Inthelimitre, thisbecomes)”, (-L"R) =u. Intheboundary-value representation ofthedecomposition components of thesolution u, Ug=Up,=Py U,Up,+U,) =O, -L"RO, 7 Cnarren 4 “1 “py U;=Up;Uy,+U;9=,~LRO, +(L"R) ©, Ug=Yom +ims te FU +Uno =,-L'RO, _,+...+(-L"R)” ©,+(-L"R)*®, and theapproximant tothesolution 9,,, isgiven bythestaggered summations. Thus, kl ke .a=2YER o,Eo’ Again inthelimit oftheapproximations wegetu;thus o-d D('Ryo, lime,=SY (-L'R)"®, =Y(-L'R)"Y ®,=D(-L'R)"@=upoe ==0 m= eet Wenow have aninitial-value format forboundary-value problems. Wecan determine ©, byevaluating anapproximation 0,., atthe boundary conditions. thenuseitintheinitial-value formatforabettersolution,ie.,one even closer tothefinal u.The two limiting forms ofourapproximation are equal Tosummarize, inaboundary-value problem wecompute 0,.,; =, ~Us and evaluate attheboundary conditions. Now wecan approximate agood value for u.from: u=02 50, Then, using theapproximation for®,calculate a=DLR) forkaslarge aswewish without further evaluations atthe boundary conditions ofo,,forhigher values oftheindex m. Dovete Decourosmiow as HOMOGENEOUS NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH GIVEN BOUNDARY CONDITIONS: Starting from theusual decomposition form Lu+Nu=0where Nuisan analytic function f(u), theusual integral representation ofthesolution u=®-L"Nu with the(single) decomposition u=7", u,and Nu=D~_, Aqisnowwritten usingdouble decomposition, u=2D Men me y=2Yas Now ourusual polynomials A,must alsobedoubly decomposed; thus: fu)= DD Aas Ei Aa=D Aas Fa Now =D toed Oe-LDYDAne a a ae where wehave decomposed theinitial, oru,term taking only u,,=, asthe first term. Itisnotessential, butwecanalso utilize theanalytic grouping parameter A.Insuchacase, >Ma,=POL-L'Y wa,a Ea} a end Ea ee Without4 Fuso-' Fa,Ea Ea 86 Coarren 4 Aninitial-value solution toLu+Nu=0isprovided by: uj=0 u=-L'A, u,=-L'A, uy=-L'A,, Aboundary-value solutionusingdecomposition oftheinitialtermgivesus u,=®, u,=0,-L"A, u,=0,-L"A, u,=0,-LA,. (There arenointegration constants implied byL'',ie.,itrepresents apure two-fold integration with noconstants.) The A,inthe boundary-value problem aredecomposed intoAgs;thus: Ao= Ava Ar=Agi +Aro Ars Agar Ayo An A=Dawes Uy=0, uj=-L"(Aoo) u,=®,-L"(Ags +Aro) B= ,-LD Agate Doves Decourosrow a7 bys=Oy LY Anse 40=Jo,-L'd D>Aas oo = == = lim® Yo,-L'Y Ya.,=0-L'Y as=u demonstrating convergence. The value of©,isdetermined byevaluating 05.1 =9q ~Yqattheboundary conditions. Intheinitial-value problem, 9,=0',u,andlime,=Di,te=uor again, O=O-L"D Ay limg,=®-L")) A,=u ‘What weaccomplish, after computing several terms of¢,,istominimize our computation forboundary-value problems byavoiding thefurther matching totheboundary conditions. Because oftherapid convergence in decomposition, computation was already minimal incomparison todiscretized methods, andtheabove procedure offers further decrease. EXAMPLES OF BOUNDARY-VALUE PROBLEMS: Wenow calculate twoboundary-value problems todemonstrate howonecan change the boundary-value format toanequivalent initial-value format, decreasing computation andaccelerating convergence byusing theconcept of double decomposition. Thefirstexample isanordinarydifferential equation. Thesecondisapartial differential equation which isconsidered both inthetemporal format (t- coordinate partial solution) and thespatial format (x-coordinate partial solution); convergence ofthespatial solution isaccelerated bytransforming it ‘nto theinitial-value format. The procedure canalso beused fornonlinear equations. Consider theordinary differential equation @u/dx?+au=B(x) co Charren4 withboundary conditions 0|oy Moose Beg [¢,=0-LY a, . c=geo ;\fime=0-1" Ag=u Blx)= Bax™ Ear} Inoperator format withL=d?/dx*,wehave Lu+ou= A(x) Then solving forLuandoperating with L",which isalways anindefinite integration forboundary-value problems, wehave u=u,-L' au where uy=Ay+Byx+ffBlx)dxdx Bywriting u=)~, u,,thesolution isdecomposed intoasumof components tobedetermined, and6,="! u,isthe“approximant” tothe solution, i.e., aA-term approximation converging touinthelimit. The one- termapproximant is9,=u,andwemusthave 4(,)= 8. O(x2)= $3 Following components avegiven by u=-L ou, and,.,=0,+u,. Theincreasingly accurate approximations must stillsatisfy theboundary conditions, hence Douste Decouposrron % dalm)=§ O:(%:)=& Gauls) = alts) =o Then for4>0, u,(u)=0 u,(x,)=0 while for 4=0, uy(m)= u,(x2)=: Since uy=Ay+B)x+L"B(x)=A,+B,x+znTcErie and weknow that (x)= S 44(82)=6: wehave F_Bax AorBintDohms =_Baxi? AoBtDdSma Let's write Ay+Bux=GI Ag+Bors =6p) where gong-F 2Ba£4 (m+l)(m+2) weg 5Be" OOSt(mei)(m+2) Then, % Charron 4 1x,) (Ax) _(&? 1ox,) (Bo) (a which wecansymbolize asasimple matrix equation xA=& orA=x" if, x,#X,,4trivial condition, since thepoints x,,xzmust bedistinct. Wenow have Ay\__ 1 (%2—m) (8 B,)x—x,(-1 1J(e Thus AyaSSP” Xo % 5eee. X27 % uyAg+Box+5Ba&(m+1)(m+2) => late, with .a=, al”=B, and a?=——Pa_ =" (m= i)(m+2) Now wecalculate theu,component togettheg:approximant, recalling that L™represents indefinite imegration. u=-L' eu, =A,+Byx~ ffoYalxdxdxEs} =gale? =A,+Bx-yAtexD Gestymad) Wehave u.!x.+=0 andu,(x;)=0 andwelet ovate Decouposmrow o Ay+Bx,=8" Ay+By, =a) and let = 0) 42 2)22 @al) xfS=Xenasd) pu ay? Oa xP=D> (m+i(m+2) Proceeding asbefore tosolve forthe“constants ofintegration” which we prefertocallmatching coefficients, ay 2 a=EGons a en _at) 3-24! x % wm gh xan?a x u,=A,+Byx-a ))—*—__&(meine?) =D alxe a withaf?=A,anda(=B,and al)=a = (m+i)(m+2) Wenow have 9,=@,+u,andcanproceed inthesame manner toageneral tem u,. u,=-L" ou, ud ale cee&(m+1)(m+2) ur(x,)= U(x) =0 A,+Bx,=89 Ap+Bay, =89 2 Charren 4 where = (on ge Bax? aay! xi">(m+1)(m+2) = (et) 4 (0)=x? ag x?Ss>>(m+1)(m+2) or (Es) (Adaehog) (Be) (ee sothat Ad\__ 1(%—™) (g/° BJxo-mt 1J(Ee We now have Ape Go x HO_210 p=22=8 xX (where, ofcourse x;,x;aredistinc’ points inaboundary-value problem) Therefore, az wallxe =A,<Bx -x7588s me x82Toh) w=Fal where ay)=A, a=B, 0,2 aa=? (m+1)(m+2) andfinally@,_,=@,+Uy. ous Decourosmow » SUMMARY: w=5al wed ae weSale and an=,Ye isthe(m+1)-term approximant tothesolution u,which wecanalso write as = = wf.) = uedFaeres Saye oSax Weobserve that (dittanh limQ..1{2a}= 2 since ved 7 & Upon substitution, where Finally wenote thatnodifficulty exists inextension tononlinear cases since it onlyrequiresuseofourA,polynomials forthenonlinear term. PT Charron4 Weagain consider thesame ordinary differential equation @u/dx? +au=(x)= >B,x™ but with the initial conditions u0)=A=CanfA}= >Ay du(0) 2SO=B=ben{B}=>By ze an{B} 2 The(m+1)-termapproximant 9,ofuyis Ga.1{Uo}= Gan{A} +%O01{B} We can now avoid further evaluations ofboundary conditions, aswedid earlier, byrecasting theproblem intotheinitial-value problem format, Wecan then continue with lesswork, ie.,without furtier matching totheboundary conditions. This acceleration ofconvergence byrecasting boundary-value problems into initial-value format becomes more helpful astheproblem complexity grows and matching boundary conditions becomes more painful, because wethen have amore accurate initial term towork with. Thus, u(0)=0,..{A} du(0)20) 61B}dx will yield identical analytical andnumerical solutions totheboundary-value format solutions. Thus beginning with Lu+au=B(x), 3 Lu=A(x)- ou usu,-L'au whereL"isthedefinite integration operator L"'=fi*f(dxdxand u,=u(0)+x20). 1Bx)dx Douste Decouposirion 9s or = a uy=A+Be+ SB _ &(m+1)(m+2) We canwrite uas uy=Dal x® s where alsa a=B = Ba ‘or? (m+1)(m+2) Wenote now that nofurther boundary condition evaluations arenecessary for computation oftheu,foranym.Continuing, u,=-L" au, u=-L' au, uy=Dagx® = = gatyebie Sea fe z° z(m+(m+2) vat Sax where - 2ea =“(m+ij(m+2) a =aal) =*(m+3)(m+4) Uggo=, ax a where a)2 aa?®"(2n+m-=-1)(2n+m) 96 Charren4 Since6,.,= 2,u,andu=YO")uu= 7,xPO,al)x®Stag- ‘gered summation isapplicable. (See Appendix TI.) SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS INSPATIAL AND TEMPORAL FORMATS: 2 2 Foee(s.nur =f(x with the initial conditions u(0,x)= T(x) au(0.x202) 560) andboundary conditions lyst)=&,(0) ulxsst)= S(t) Wesuppose oandBaregiven intheform: Bst)= FY Baax andtheconditicins arealsoinseriesform: u(0,x)= D7) x* Ea 9u(0,x) _S22) =us)ac u(x,t) 20 Es utl= >oe LaL=@ /dx* andwrite Dovate Decouposimon 97 Lu=A(x,t)- a(x,t)u-d*u/ae? usu, -L'a(x,thu-L(a"/ae*ju where u,=Ag(t) +xBy(t)+L" (x,t) Thesolution uisthedecomposition u=J”,u,andtheapproximant is =D2u,.Now 9=Uy (xt) =60) (st)=8.10) ‘Thegeneral component is uy=-L" @(x,tju,,-b(a/ae ja, L"()=A(t)+xB,(0)+ff(dxax Clearly Gast =Oq+ Ga(Rst)=Git) Ga(Xaot)=S(t) Gan(Xt)=5(0) Gani(%rt)=&(0) Thus for m>0, u,(x,,t)}=0 ug(x,t)=0 and form=0, us(xist)=E(t) Uo(Xast)=S(t) We have 98 Charron4 uy=A,(t)+xBA(t)+ff B(x.t)dxdx Atsst)= 5B(x Balt) Bas ® Wecanalsowrite A(=y Alle B,(t)=xBO Now mAs 5felt)? u,=Ag(t):xB,()+ Ta+D@ey Wenow have thefirst approximant; itmust satisfy theboundary conditions hence, LS Belts? Ag(t)+%,Bo(t)+zcesyersiia E(t) yee alte? AdsBOD Tejas a) Let's write thisasAolt)+%Bo(t)=5°(0) Ag(t) +x,By(t)= &(0) where 29 esq 5 —Baltar?Se(Y=S(t)=(m=1(m>d) 20 aya 5 alt?SP(=S(t)Dy@=)(im>d) or (1x)(Ao(t))_(20) taas)(Bec) Lev Dousis Decourosmnan 99 from which wefind that Ag(ty=SW =X, supe SIE XTX ‘sothat =A+ $_Bolte" =A+KBO+D SA ‘becomes By=Falias where= a= Ad(t) a(t)=BC) and a),(t)=Bald orth!(m+i)(m+2) Since A= at B=5Be Balt)= BSL wecanwrite PAE Meal ae WHT We=y woe o Wye¥ = Bast” Wee a, ve as - aa()=2aaat“Zz(m+iime2) 1s wedzalx=o, or 100 Charren4 a=¥a(yx? where“ a= 2aet* Wecannow calculate theu,component andthe¢,=@;+, approximant. (We point outthat although weexplain inconsiderable detail, theprocedure is simple andstraightforward andiseasily programmed oreven calculated by hand.) Fortheu,component, wehave u,=-L? @(x,t)u, -L(a? 90ju, aa.tun(e={F 50,xhEEaxt" which istheCauchy product a(x,t)u,(x,t). (See Appendix III.) L*()= A\()+xB(1)+ ffaxa A(p=y Ae Bey Bor u,=Al)=xB(-ff DYDee YY ayas Hildxdx =fDY Car1in+2/a,., xBaxdx ced Douste Decouposmow 101 =A,(t)+xB(t) =~.el(osnin=2the+ ++GagayaL,>) mam +2) Since u,(x,,t) andu,(x,,t) must bezero AQ+xB(D=F(0) A(t)xB(Q=80, where oar |~= (ns1in+2)Qe+LYGanevAe| PMany Dre (m=i(m+2) ww [CFV+2)DSOenaw8 D() = x2 = BePOY Dee (a+im+2) 1x] FAC] _/EP(t)F*}B())(2) . sothatweobtain a=B= M0 XX By=20-20‘XX (Ofcourse, x,#x,.)Wecannow write w=FEMor where~ a)()=A,(t) a}(1)=B,(t) and 102 Charren4 oO @Me-Fe mosala (Oe-Be (m+i(m+2) Note that A(d= ale a BYt)=>Bye Finally= wD Dae =o +u; where ad=al? a= 3, [enn +2). 2+DDOana tl Bese (m+ij(m=2) 7 or wed or where- a= aes a Wecancontinueinthismannertocalculateus,us,...Tocomputethegeneralterm u,and the¢., approximant, wewrite u,=Lta(x,tju,,-L(2?/ae? ju, aD Daa a=xzy(n=1)(n+2)alyx® > as agv= >Fatt Dovnte Décourosmiow 103 TheCauchyproductax,t)u,.;(x,t) isgivenby L*()=A(t)+xB,(1)+ffdxdx Atoed ave BA(t)=yBye w=A(oesBd)-ff ETareEYaeuals eax -fj>by(n+1)(n+2)alQ), x*efdxdx uy=A(t)+3B,(t) we,[ormorantth+SSasaals” whee (m+i)(m+2) u,(x,,t) =0 u(x,t)=0 Let Adt)+xB(t)=S() Adt)+xB,(0)=52(0) where sax =ee CESCES) 104 Charron4 --[in-mnn-2ya HDSseeai|s-ny Letam) imea1a-XONMESZO) Consequently, toryan 219 Ay=SOE)=X aioe49-Ea We can now write° : ueDal (x® with~ al!(t)= A,(t) a"()=B,(0) r 7 7 eM 2A+YYanne2”| alsWad 7 (m+iy(m+2) Since Adt=y AD Bt)=xBO Therefore,“ Oy 0+ where a= alo a<3 Donte Decourosmow 10s Joyoo+2+ Sasanaf? Savas (m=i(m+2) ‘We can write theresultas w= a(x® s ane Sae 4 SUMMARY: wed Fae wad Feet aa Theapproximant 4,.,{u}= Dr,uy Lae ao}=y YY Qe BSS Bole =DDYPebfaach as Wenotethatfim4,.,{u}=u andthatlim4,.,{an.}=ae.- Since usD7, wand wed Pager ‘Substitution leadsto— 106 cuore4 uP Dae =DE axe wherea=, as TRANSFORMATION OF SPATIAL SOLUTION: Having calculated u,,weknow that .n{AD}= >Arlt) a 2,..{BO}= 2B,(t) 6,-1{¥}= {Al} x0,.-{B(0} We can now recast the problem into aninitial-value problem format, accelerating convergence bysimplified further calculation through avoidance of further boundary condition evaluations: Sesa(stus ZS=purr) oe a u(t.0)= A(t) 2x9) «3(0) ox Weproceed todesired accuracy bychoosing A. u(t.0)= 65..{A(0)} du(t,0) 2060)gy1B ae BO} Dovste Decourosmiow 107 z apeLO=F5)andLo=ff ()dxdx.Weemphasize thatL?now represents definite integration. Upon substitution, wehave Lu=B(x,t)- a(x,tu-(#/ae)u wsuy—Lt a(x.tju-L(97/90)u where. vy=u(t,0)x28) 61B(x.1) ax 5_Ba(tx** =A(t)=xB()= 5Bal ta=A()*xB(0)z(m+(m+2) with A= yAt B=) Be Therefore wecan writeuyas ” uy=D alo) x® where~ a(t)=AC) al(t)=B(t) 9(t)2 Balt)aa) mad)Wecanalsowrite w=>DYalee withms al(= Ay an()=B, 9),()=Pas _ aetaal=med) Tocalculate themthcomponent u,ofthedecomposition ofu,wehave ue=-L" a(x,thu,., -L(F/at Jug, 108 Charron4 whereL"()= f"f°(Jaxdx.Thus, u,=-L"a(x,t)u-L"(97/dt)uy mit (#/a0*)u, =¥y(n+1)(n +2)a@)_,xB" agns)ustnd= SEx Sasa tl Be met sothat =- nea psdeberdwae Be ee) SS ortysMFNin+es Ee etm) where “eeis 22 7 TCOMO=2eE=DNacransBe Bee= (m+ Iim+2)~ with mat 7OFMOn2DBereanHe| Bae= (m+iy(m+2) ‘Theapproximants o,.,=),_, u,andu=7,u,arecomputed asusualso that Doumts Decourosmow 109 Staggered summation canbeapplied atthispoint. TEMPORAL FORMAT: Consider thesame partial differential equation 2 2FeeabssusZ2=p00.) with theinitial conditions: u(,0)=1408)=ay)x du(x,0) eecane ie)x! ‘andboundary conditions: oGt) == are uast)=4)= >gre ‘Wewillsuppose that@andiaregivenintheform ax.t)= >Ya. xtt cad Blxt)= YDBas =D B(x) me Ea where B(x)= DYBawx Toderive the solution wedefine asusual L=d*/dt? and LQ=fi()dtdt, a(two-fold) definite integration operator. Then Luta(s,t)u+(2°/2x7)u =B(x,t) Lu=A(x,t)—a(x,t)u—(2°/ax")u andoperating with L”weget 10 Courren 4 usu, —Lo(x,thu-L"(0°/dx"*)u where waaretaQd+f fiAltera Solving bydecomposition, thesolution uiswritten u=07,u,andthe approximant 9,=J.)u,.Then = Up O44, =O +Uy up=-L" @(x,thu,,-L'(a*/ax?)u,, Now wecalculate theuscomponent (and 9,approximant) ug=7(x)+t72(x)+L"Bla.t) . S B(x?weno) DYBrean h andwenotethatwecanwrite a(x)=>Tlx® ot B= >Bx) Bos)DBaex® and- uy=,a(x where “ ayi(x)= T(x) a)"(x)= 22(x) 29,()= Bb)— (n+i)(n+2) and Dounuz Decourosiiow m1 ae? a)=—Bea =a? (n+1)(n +2)Thuswewrite = DDalt=9, uy=Faayet where: = al(n)= Dae x a Now wecancompute theu,component and9;approximant: u,=-L?a(x,t)uy -L(?/3x7)u, 2en Su,=¥Caenmernd,. xe an=S Faget | me TheCauchy product 0(x,t)ug(x,t) iscomputed as atone)={EFane‘|{E52e| (which canreadily beprogrammed). -fifLE(m+1)(m+2)a%,., x®dtdt m Cnaoren 4 =,(menimsamttireS Seriesatt| =-teelTeg werkEee (a+ ijn+2) whichwecanfinallywriteas «aS wetDYal(xy with= alix)= Dall x* We now have @.= 4,+u,and can continue inthesame manner totheu, component andthe@,.,approximant. u,=-L a(x.tju,,-L(8?/ax? ju, am Aru LE(rsew-anhare aixt)=>Saxe Wenextcompute theCauchy product indicated bya(x.t)u,.,(x.t) oF Sean wef fvoy xx®eyYDeecpane wh”atdt -ffcoy +x™(m41)(m+ 2a,dtdt a= SY ea’, where -|(m+1(m +2) +Dewanean] ale =) v0 Ji= ee (2¢+n-1)(22+n) Pry =O +Uy SUMMARY: wed yaar wae Feet a 201=D8=darfu} MoD OY Yaloe Staggered summation cannow beapplied. COMPATIBILITY OFEQUATION AND CONDITIONS: Ifthecomputed solution satisfies theequation andtheconditions, wehave thesolution. Ifthephysical problem iscorrectly modelled, nodifficulty appears. One cannot arbitrarily assign conditions toanequation. The equations, conditions, andsolution must beconsistent. Ifattempts tomodel a physical system failtogive uscorrect conditions, one cangetboth temporal andspatial solutions andfind that these solutions aredifferent. Ifthey are close over afinite region, then werealize themodelling needs improvement. This may allow ustodevelop apredictor-corrector methodology which we leave tofuture work. 4 Cuarren 4 Ifincorrect conditions areused forthedecomposition solution, wedonot have asolution. Ifthesolution iscorrect, i.e.,itsatisfied theequation, wecan doinverse operation, e.g., inLu+Nu=g,wehave u=@—-L" Nuandcan solveforF.IfLissecondorder,weknowuy=-Lgwhere®=a+Bx; hence, weknow Fand have atestfora,b. NONLINEAR BOUNDARY-VALUE PROBLEMS: Wehave two alternative, actually equivalent, approaches forboundary-value problems, whether ordinary orpartial differential equations areinvolved. The first istomatch each approximant 9,forn=1,2....,n totheboundary condi- tions. Inboundary-value problems modelled byordinary linear differential equations, only one such matching isnecessary. Wecan carry along theun- evaluated initial term without evaluating theintegration constants bymatching totheboundary conditions and only dothematching when them-term approximant hasbeen calculated. Innonlinear differential equations or(linear ornonlinear) partial differential equations, this isnot possible. Then the matching must bedone foreach level ofapproximation. The second (ordouble decomposition) procedure adds decomposition tothe initial term. This allows aconvenient match totheboundary conditions ofthe approximant 0_because theconstants from each integration areadded togive abetter initial term. Aswewill discover later, the solution can then becarried further, ifmore accuracy isneeded, asaninitial-value problem. The value of decomposition oftheinitial term isinthematching oftheboundary conditions byadding alltheintegration constants C,,,and¢,.separately toform anew cp andc,andnew initial term which isnow close toafinal value asn~~in@,. Now wecanusethis upterm without adding further constants ofintegration and matching toboundary conditions, since forhigh approximants toachieve accurate solutions, thecomputation ismuch less. REFERENCE 1, G.Adomian and R.Rach. Analytic Solution ofNonlinear Boundary-value Problems in Several Dumensions, J.Math. Anal. and Applic.. 173. (118-137) (March 1993). CHAPTER 5 MODIFIED DECOMPOSITION The modelling ofphysical problems can lead toordinary orpartial differential equations which arequite generally nonlinear. Examples include equations such astheNavier-Stokes equations influid mechanics. theLane- Emden equation forstellar structure, nonlinear Schrédinger equations in quantum theory, soliton equations, etc. Wepresent here avariation ofthedecomposition method which canalsobe applied tosuch equations toobtain accurate quantitative solutions. A mathematical advantage ofthevarious adaptations ofdecomposition isthat linear equations areaneasily solved special case andordinary differential equations areaspecial case ofthetheory forpartial differential equations. so wehave asingle unified field. This alternative formulation will bereferred to as“modified decomposition” [1,2]. Itrequires thefollowing result on transformation ofseries [3]. Normally wewritef(u)=)”, A,(Uo...uu,). However, givenaconver- gentseries u=S\~, c,x*andf(u),wecanwrite F(u)=D) XA,(Coyne) Ea} We can see this asfollows. Let w= xt BS Wewishtofindatransformed seriesf(u)=*(Dz.6%")- Since f(u)=2,Ay(Yost), flu)canalsobewritten as DYAalCoveesCa)e™ ‘Thus forf(u)=u’,forexample,we have lis 16 Cuarren 5 Ao(Uo)=us A,(up,u,) =2ugu, A,=u}+2u,u, Since Uy=Cy,Uy=CX,Uy=C3X7,..., Wehave £0) =DAg(Cooney)x" FE with A,=c3, A,=2c,¢,,.... ASanexample consider tan”x andf(u)=u? tan"x=x-x1/3+x3/5 Ap=up=x? A,=2u,u,=-2x4/3 (tan"'x)' =x-2x4/3+- Power series solutions oflinear homogeneous differential equations in initial-value problems vield simple recurrence relations forthecoefficients but generally arenotadequate fornonlinear equations Dealing, forexample, with asimple linear inhomogeneous case Lu+Ru= g withsecondorderL.welet e-Dax ura,+ax+ >g.x°- plu where Jisanintegration andI’willmean atwo-fold integration. Thus DYox*= a.-ax+>)gx°?/(n+1)(n+2)- pyc,x*7/(n=1)(n+2) with¢,=and;=oj,Forn22 Moviteo Décourosmon 7 ¢,=BaiaPoaen(n 1) soweobtain coefficients from arecursion formula, The technique provides an interesting alternative forequations such astheDuffing equation and theVan derPolequation. ONE-DIMENSIONAL CASE: Consider the nonlinear inhomogeneous ordinary differential equation Lu+Ru+Nu=g where Lu=d*/dt,R=p(t), Nu=a(t)f(u). Wewill view this asaspecial case inone dimension ofamulti-dimensional partial differential equation. (Inthefollowing sections, wewillconsider equations in wo. three, and four dimensions.) We can write uD ae Fed R=p(t)= >pt g=a(t)=> gt a-Dac Ferd £2DAx(Q0a,)P= >AL S Ford u=0,+L;'g-L)Ru-L) Nu where=+4,andLy’=f/f’()dtdt.Thesubstitution yields >ata+tr+ffDygraa-f'f {5Pyv}{5a.fova -ff{5ae1SA,voce Multiplying andcollecting likepowers oft, ns Coarren 5 and Replacing theabove quantities inbrackets with theequivalent expressions on theright side, xatatrins ff>gat?dtdt feosAEeSona,fvaBS onfSae ) -ff5eya,Acafatdt Carrying outtheintegrations, wehave =. = oeavertin+y —se, RaveneneS Goyer & =~wt -Y Spa2Ganmsy yet -Y yaa2Daneey Zohn Inthesummations ontheright,ncanbereplaced byn—2towrite - = oe vencn+y —e. =at=t)7tt,zmeant iona-)aQr = oP oe ee ea Finally, wecanequate coefficients oflike powers oftontheleftsideandon theright side toarrive atrecurrence relations forthecoefficients. Thus MooirizoDécourosirion 119 a=% ast and forn >2 B.D PebeneDyOeAcre n(n—l) This solution, ofcourse, is u()= ae Ft TWO-DIMENSIONAL CASE—NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS: Lu+L,u+Ru+Nu=g Wenow have two linear operators Land L,.LetL,=d*/at? and L,=0°/dx*, Assume thatwecanwrite u=d ayes BG or = LS Fe} Ifwehave u(xy)=~_,DagCuaxX”y*,wegetterms 0.0» CoYrCoaYreerGyoKoCyXY»C2XY"voee9Cn9X" 9CryKYse Thefirstgroup canbewritten x?” cy,y®,thesecond group as xr, CeYsthethirdgroupasx*J)",c,,y°etc.Thus u(xy)= Dca(y)x* a where c,(y)= D,Coa¥”Sothatthedouble seriesiscollapsed intoasingle series. Wesuppose theoperator Ris 120 Cnarren 5 R=p(tx)= YY pat xt ied ~T= 2 “2d[Breafdpoor Letthenonlinear term Nu=ai(t,x) f(u) and a(x)SFa8=>“E.3'|-E a,(x)t* init mole a Writef(u)=7, tA,(ao(x)--.84(8)) =Oo,MA,(x).Let -= -fe Leseein DZgata ZlYeex]dals) dec}Fa tert fod Thedecomposition solutionusingthetpartialsolutionisgivenby: u=0,-L'g-L L,u-L?Ru-L'Nu where ©,=F9(x)=10,(x)=u(t=0.x)+t5u/At(t=0,x) andL;'=f'[/()dtdt. Substituting foru,f(u),g.andp,wehave >»a(x’=2,(x)+14(x)~ ff)xyg,(x)t"dtdt -ff@rax)E aoeaa ff‘5plx)t?>bya(x)}didt anle = ) ~ff>eaten}=As(a)ejava The bracketed products are sooinen becourostron 221 t= = ) ee[Soe|{3stave EeEa...t0 byassy}byagoe}-&Cby(x)A...(X) Substituting theabove products by(xe=4,(x)+10(x)+ ff)byg,(x)t°dtdt -ffiwon a,(x)t*atat} wee [om .“LLCS 2cepa,to)ae -{i{5EDaA,..(a) att Wenow carry outtheabove integrations towrite ~ oa Lacon) en) Soar HO) 3oa S0&%(n+1)(n+2) ax * Penner STansy &Pe) aeeeanan ZSTaney &MMA Let n>n~2 ontheright side, Then za(x)?=T(x)+0+,cc85-2(8) - os F“2ieee te 122 cuwpren s = op &sary Pelt) oo ~Y FY a0) gal&rary MOA) Finally,equatingcoefficients oflikepowersoft,wederivetherecursion formula forthecoefficients (x)= T(x) a,(x)= 50) and forn >2, 8-203) ~(8/9")a, 208) D{0,8} 2-8)+(RAL.62} 8,8)$$ n(n~1) ‘Thefinalsolution isnowgivenbyu(t,x)= .__,a,(x)t*. (Whether modified orregular decomposition isused. wecan still apply Padé approximants, Shanks, Wynn, Euler. orVan Wijngaarden transforms toaccelerate convergence.) THREE-DIMENSIONAL CASE—NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS: Consider Lu-i,u-L,u+Ru-Nu=g where weletL=ovat’. L,=o sox", Ly=d'/dy*, R=p(tx.y), Nu=a(tx.y){(u), We now assume wed YYaen'y’ “Leyy aory’2Dale) R=p(uxy)= >YYPaty! =Yey YSausty =Dpleyy “MoontaDecouwroy a=a(tx9)= 3DyEasy -Efzzauonty|-E a(xy)t" )=3PA(a(x).-ae)= Zea) “5ap»tasty|eEUg(xy) The tpartial solution is hiuso,+L)g-LLLyLuLyRu-LNu Wecannowwrite Dya,(xy)P=T(ny)+ta (xy) “Ll.Ztas(us)ata Karax)S,ta,(x,y)dtde LiEron| {Sesoo9]aca ‘Thebracketed quantities are sae cure5 which wecan substitute toobtain Eee ralsy)+15(x.y)ffErg,(xy)dtde . SL2eZeasy Lear Zealxraa EOE @,(xy)An.(ova which wenow integrate toget ores) =ay(x.y) it PKYa(RY) Wenow replace nby n—2ontheright toget Mooinen Decounosmon ns Sea(a)mnley)eler)+ Dssale) ao =~ 3Dery agLeagan ayea) edZar PLy)a,24(%¥) ~ oe oeeran 206,(5Y)Ag2-o(%Y) Byequating coefficients oflikepowers oft,wehave a,(x,y) =7,(x.y), a,(x,y)=7,(x,y) andforn>2, a,(xy)={8,-2(%y) (2°/2x")a,_(ny)(2/2y")a,(69) aS[plsydassoley)+a,(uy)A, 2-609)foto Fo Thefinalsolution isnowgivenbyu(t,x,y)=S, a,(X.Y) FOUR-DIMENSIONAL CASE—NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS: . Lu+Lu+Lju+Lu+Ru+Nu=g where weletL,=2/90, L,=2°/8x7, L,=3*/ay', L,=F/d2*. We can assume w= DD tect tyzt Baa Bla S J =Yr'a,(xy.2) B 126 Curren $ R=plny2)= DY DD Parent xyz Bane =F EEEyeteasee |Erato Blame1 Nu=a(t,x,y,2)f(u) a=VLD Lewy 2?ete Bam ms mls Fad f(u)=JA,(a9(xy,2),--0a,(%9.2))= DMA(YZ) exatuxy2)= >OES Usty'2 acemee Ss HLL LDVvyeeve HdCeey.z) The tequationis u=@,+Li'e-Li[L, +L,+L,Jub’Ru-LNu with ©,=ult=0.x,y.z)-t du/dt(t=0,x,y.2) =t)(x.y.2) +t7,(x,y.2) and wecanproceed asbefore with substitutions andintegrauons tgetrecursion formulas for the coefficients. REMARK: Wehave seen thatnonlinear partial differential equations aresolv- able bythemodified decomposition procedure using concepts ofthedecom- Position method (partial solutions, theA,and transformationsofseriesusing theA,,). Wehave seen previously thatsuch equations aresolvable bystraight- forward decomposition also, Comparisons cannow bemade; ingeneral, de- composition solutions converge faster butthesolution isidentical. Asimple example isL.u+L,u+f(u)=0 where L,=0/dt, L,=0/dx, f(u)=u", u(t=0,x)=1/2x. Thetpanialsolutionis u=,-L;' L.u-L} fu) Mooiteo Décourosmow : oe where ®,=u(t=0)=1/2 x.Then u,=-L;'(0/dx)u,- LyAy u,=-L;(2/2x)u,-L? A, wheretheA,aredefined foru®.Wegetthe(decomposition) solution et feteHy.)uagbegtagt J 1"hoifet ‘Using modified decomposition, let u=5a(x) Nu=5A(x) hence~ - % -_ Y8A(x)= t(x)-|YO@/Ax)a,(x) at -f»eyayao(s)at~fzeyA,_,(x)dt ¥8ag(x)= Hx)-F(07/041) (9/dx)a,(9) “Tle +S aoo-F(e7m+)S ALG) Efa,=2-F(/n)(a/9x)a,5 fromwhich weget soa! 128 Charron 5 a=tH=1/2x a,=-(9/9x)ay~~Aalay) a,=-(9/4x)a,5(1/ Ma,=49)(12XA,~Ap) =1/2x?~(1/2)(1/4x=1/2x)=(1/2)(A, Ao NoUraylyae1/23)172¥As~As)sothat ve wedtee]2x| 2x ae | which isthesame solution obtained bydecomposition. For homogeneous equations with aconstant coefficient off(u) asinthisexample, therateof convergence isequal. For thegeneral case, wefind that thedecomposition method converges faster. Suppose weconsider thesolution ofanonlinear equation where the nonlinearity andtheexcitation aregiven asapower series, orequivalently as graphs from which series arederived bycurve-fitting techniques. Let's write u and f(u) inseries form u=d ae fu)= >,au" fu)=>,aSast: ise gpl Computing fF,a,t}weseethat {Zac}-Eeadens forall2andforv>0,(Ofcourse,forv=0thequantity ontheleftisclearly equal to1.)Thus, ifwedefine Moviriep Decourosmrion 129 = 7) by=O+>&AguIfa, my b,->a,Auas forn>0 ‘The A,areeasily evaluated. Forexample, A,[u’]=u3 A[u’}=vu,uy7 Weobserve thatA,[u’*']=u,, ic,thelinearcase,then ALCf a,=AsLe”Icnn, =Bales) The previous resultontransformation ofseriesstatedthatifu=J)”,a,t°, fW)=Y PALM, From (1)_ bo=oty(1)+a%(a9)+0,(a5)+%4(ag)+...+On(a9)+.. be=¥a4(23) b,=@,(2,)+a,(2aya,) +0,(3aja,) +--+a.,(ma,az) b= @,(naay") by=&%(1)+(a)+@,(a5)+a%,(ag)+...+o,(az)+... bo=>a,(a3) CTS4,=a5(0,)rc(2a40,)+a5(30h)++cay9") b=) a,(na,a5") 130 Charren 5 The a,dominate theconvergence. Forsimplicity ofnotation,let AQ)=ALUfayen, =As(Borns) Then bo=G+>a,ADa)= >a,a5 = = b,= a,AM"(ay,a,) b= @AD(aayaz) = be=Dd,&AL(age) fw=¥ bth=a,+ >@,Aa) 1, @,AMap.a,)= 8S a,AM(ac.aas) aUY@,AM(aya;ayay)oe where= f(u)= a+ YY a,AM(ap.....a,) mF f(u)= >bv" ifusDO,atands(u)= 7,@,u" THEOREM: IfS”,at"andf(u)= 0,au’areconvergent, then f(u)=07,btisconvergent where MopirteD Decourosiion 2 b=a+), &Ad(U’)farea boo=2&Aa(tlaren, This isuseful inthe following problem. Consider (using modified decomposition) thenonlinear equation, Pu/de +Nu=g(t)=> gt" withu(0) =c,andu’(0)=¢, andNu=x,au* a.wry ou-zgat Substituting =D at du_< => (n+l)aexe2. ysPBF(nen+2)a,.,t°aS andusing thetheorem above Y”,a,u*=\~, b,t*withthegiven formulas forbs, Y(a+Hr+2a, 0+)b= >ge Ed Fa card Letg,=(n+1)(n +2)a,,, +b,sothatwehave found thecoefficients, B= Cy aac, 8,—b, a,= See (n+ 1)(n+2) 132 Charren 5 withu=0",a,t.(Ofcourse,wecansolvetheproblem bydecomposition awell asbymodified decomposition.) Letusconsider, asageneric example, apartial differential equation inthe operator form ofthedecomposition method: Lu+Lju+Nu=0 where L,isalinear differential with respect tox,Lyisalinear differential with respect toy,andNuisanonlinear term. Thex-dimension partial solution is u=,-L3{Lju+Nu} where L,©,=0,Assumethesolutionintheform uxy)= dDaaaxy” me =) &(y)x* where 2,=", ay.y®.WritealsoNu=f(u)= 7, A,(y)x™ where the A,(y)=Ag(E:(y)----.$e(y)) areourpolynomials. Substituting. wehave &sel =O,-LLY Sy -LIL, DSAa(yx® sine7 Fr = LEY Saly)x* =DSaly)x*7/(m+ (m=2) =LS_-2(y)x* /m(m-1) at and LD A(x =X Agaly)x®/m(m=1) we can write~ — &Stow"=9,-FbSalone/mim—9-¥ Agstoye*/mim—) Mopirie>Decouposimion 13 Assuming L,isasecond-order differential operator, 9,=koly)+xk,(y) where thekoandk,can bedetermined from thegiven conditions. We finally getrecurrence relations forthecoefficients &,, Sal) =koly) Sly) =ki(y) and form2>2 Salv)={bySa-sl¥)*Aanaly)}/m(m—1) where Ag(y)=Ag(So(y)--a(¥)). They-dimension partial solution is similarly obtained. Assuming L=#lay* u(xy)=m(x)¥* Ford f(u)=>Ay"E4 where Ax(x)=A,(Mo(*)---.Ta(%)). Ymy" =,-LLYn(xy*-L7 YA(xy®Fd far Ea} Proceeding asforthexpartial solution, wenow gettherecurrence relation ox)=¢o(x)n(x) =6,(x) and forn >2 g(x)={LaMya)+Ay-a(3)}/a(n 1) where A(x) =A,(19(x),---+M.(%)). 134 Charter 5 REMARK: When kj(y) andk,(y) areexpressed aspower series iny,and Co(x) and¢,(x) areexpressed inpower series inx,theseries solutions u= (yx = =D n(x)y" Fd can bewritten as uD beexty Consider asanexample theequation Fufax’ +au/dy?+plx.yu+ a(x.y)i(u) =g(x,y) Since wehave chosen atwo-dimensional case with theconditions, du(0,3i : u(0,y)=Cy(y) and> Cy) x ‘we assume the solution inthe form and similarly write—— P= DPear¥* w= me (Anordinary differential equation suchasdu/d x"—p(x)u +a(x)f(u)=g(x) becomes aspecial case asdolinear cases ofboth theordinary and partial differential equations. Weuseasingle series, Forathree-dimensional series, we use atriple series.) We write the differentia) equation inour usual decomposition form as Lyu+Lju+Ru+Nu=¢ This coule also tesolved with noundary conditions. Mooitie> Decourosirion 1s Wecanuseeither thexortheypartial solution. Using thexpartial solution for which wehave stated conditions, weoperate onboth sides withL;’towrite (0.tet ““ LiLyusu- (0.y)=x2400.Lig-L}Lju-Ly Ru-LNux where L7!isthetwo-fold definite integration from 0tox.Wenowhave u=Cy(y)+xC,(y) +L;g-LyLu-L; Ru-LyNu Computing L;!gweget List) Dtay =D Deax tylnti(n+2) =DDBerar?y*/n(n-1) AS Computing L,u, wehave Lue(Hav)E Eee aa. =DZYmm-Ne..xy** mam =XY(m+(m+2) cay am Nowcomputing Ly’L,uwehave LILw=L YDDY(m+i(m+2)e02%°¥* am =F F(m+D(m+2) aeye“ZZ Ganon eee =F F(e(m+2) aye>»Dyn(n=I)Cynara ®¥ 136 Coarren 5 Computing Ru, po=ass}{5=cary} Bm een] Fier Corde Next, Foire Corder} Also «=f¥ Saevy)-¥ Facey mt mm where thepolynomials A,..(Co.:--++Co) areobtained bywansformation of series, We have ae \ia= nuratiul=(3Faax'y"){5yay}BS Ils } Then Li!Nu=>>DaYa,Aven[ineineayst"Salam i} -EE(SSe, Aesioohty at am [Se WealsowriteCx(y)=S), Coa¥®andC(y)= 7,Cin¥*.Weknow 5,94€o,»sCo,>.Similarly, weKNOWC).5,C,. Cy.g-05 u=C,(y)+xC,(y)+Ly g-LyLju-Ly Ru-Ly Nu and we can write Movin Dscoupesmon 137 ny a_¥ = = ay FS Bete yyeDV aa y=dcoy xd car tdYextyam Ed = &S&Snln-1) SS (m+1)(m+2) oye - oe las= ey ee -=(te 1LL LDeaster eaeu/Mn- Dpy™stash (oa J ao fst } “ZZ EL derneelmore Estee ocho} Wenow equate coefficients oflikepowers. Weareusing thexpartial solution; hence, weareparticularly interested inpowers ofx For n=0, Cra =Coa Forn=l G4=Cig Forn>2,wehaveforlikepowersofy®arecurrence relationyielding the coefficients go Bene _(mtI(m+2) | sce ma) 2 a 3PowSs-ronmea FSCogAsctnmesn &zn(n=1) >»»»n(n=1) andcannowwriteu(x,y)=.-, Dinwy CamXY"sincethecoefficients are determined and theA,canbefound. Since Cy», Cy:.C)3 areKnown by decomposition ofCo(y) wecanfindother components, ¢.g., ¢,,,depends on 3.Similarly C,(y) yields components c,,,forallm,soc,,,forexample, is found from ¢,. The linear cases (a=0)areconsiderably easier since theA,become unnecessary. Also therecurrence relation simplifies ifg=0or p=0.For example, ifweconsider theequation 9*u/ dx?+d*u/ dy*=0wehave n(n=1) . 138 Charron 5 MODIFIED DECOMPOSITION SERIES AND BOUNDARY-VALUE PROBLEMS: The “modified decomposition” series solutions have been found forinitial- value problems byincorporating andadapting ideas ofthedecomposition method. Now using thedouble decomposition technique discussed inrecent Papers, theprocedure canbefurther generalized toweat initial-value and boundary-value problems inasimilarandcomputationally efficient formulation with anacceleration ofconvergence. We will consider some progressively more complicated problems. LINEAR (HOMOGENEOUS) ORDINARY DIFFERENTIAL EQUATIONS: Consider theexample d’u/dx?+pu=0 forDirichlet conditions u(x=&)=b, fori=1,2.Letpbeaconstant (tosimplify thediscussion) and seek the solution inthe form ofaMaclaurin series usDax = w=Du, u,=a,x" Inthe usual operator form fordecomposition solutions, this equation is written Lu+Ru=0where, inthiscase, L=d°/dx? and R=p. (Of course, themethod was developed formore general equations.) Now write L"Lu=-L'Ru where L”isatwo-fold indefinite integration yielding u¢,—¢,x; hence wad ax=cy=xe,-pff Ya,x"dxdx =,+xc,—p>at[neyn=2) Ed =e,+x0,-p >a8foby) Est Equating coefficients, a,=c,anda,=c,.Forn>2wehavetherecurrence relation Mooi Decomposition 139 a,=-pa,_; /n(n~1) Using double decomposition, =yc a g=D a= a Alsoa=c\"’, a\®)=c\"', andforn>2,a®) =—pa'!/n(n-1), achiev- ingadecomposition oftherecurrence relation u=YuaD Lowry Due ie. - = ut) bystaggered summation. (See Appendix I.)Weusethestaggered summation torearrange thedouble decomposition components ofuinto anew series to achieve adecomposition suitable forboundary-value problems. (The switching ofnand mmakes theresult computable; only thesum matters — thecomponents arenotunique.) ‘Wemust nowdetermine c\")andc\*)byuseoftheboundary conditions. Insteadofu(x=,)=b, andu(x=,)=b,, weusetheapproximant Garito thecorrect solution u.Wecanwrite thisas@,,,{u}, anoperator onu. Similarly, theapproximant toa,is et oa{a.}= >at? fod Then $,{20}= as? oi{ay} =a? 1 Wewill have determined thesolution ifwecanalso compute thevalues of cl") and cl)matching the(approximant tothe)solution withtheboundary 140 Charen 5 conditions Gaar{u}(x= §,)=b, Gnu{u}(x= S2)= bs Forthestaggered series ofu uu,=al?+ax uaallacral eae? uy=a +alx-eallx?eax?alateala? tag=ale+alhealex2)+al)x2? Equivalently. rast . ug=alee where [n/2] isdefined asthegreatest integer value lessthan n/2. Thus the staggered summation hasresulted inadifferent decomposition ofusuitable for boundary-value problems. ~ = te! . REMARK: The greatest integer function used here isforasecond-order equation. Forathird-order equation, wewillhave [n/3)andforfourth-order. weuse[n/4]. Next, wederive theapproximant forthestaggered series ofthe solution; thus@,,_,{u} isgiven by: o,.fu=u,+u,t-+u,= >)asx>alex? Sat aFasnaxtahex!a, 05-{U}=Gar{o}+Xn0y{8)}+XOmes{22} 27°g {as} +--+27°0,{a1g} +O°0,fas0.1} orequivalently Morrie Decourosrion at ren anit}=YiPanraras{25} Weusetheapproximation oftheboundary conditions Paaiful(x=§)=d, Gaur{ul(x= §,)=b; inorder tocompute theconstants c{*)andc\®!andconsequently thecompo- nentsa”)anda{*)oftheMaclaurin seriesforthesolution, thusdetermining the solution u. SUMMARY: The basic steps are: 1)Compute a{*)anda{*)bymatching thesolution approximants tothe boundaries, i-e., Gaalul(x=6)=by Ganful(x=f,)= by 2)Using therecurrence relations forn>2,ic.,a=-pal®) /n(n-1), compute more components a‘")toimprove theaccuracy ofthesolution approximants. Thus wehave - a> ao and therefore w=Dae a satisfying boththeequation d*u/ dx’+pu=0 andtheconditions ulx=6,)=b, u(x=6,)=b; Wecannow accelerate convergence bygoing toaninitial-value formatted solutionwithoutfurthermatching ofsolution toboundary conditions. Let 142 Charron 5 8)=Onaif{ao} 31=beufai} and forn>2 a,=—p ag/n(n—1) which gives usanew andimproved uytostart asaninitial-value problem Genr{U8)}= 2nif80}+O0.rf8i}x whereu{?istheapproximate initialvalue.Nowu=.~, u,=~,a,x* NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS: Consider asaspecific example d*u/ dx?+@f(u)=0 given theconditions u(x=é)=b, u(x=é)=b; WeseckMaclaurin seriessolution u=J)”,a,x".Theequation inoperator form isLu+Nu=0with L=d?/dx? and Nu=cf(u). Operating with L". wehave u=cy+xc,— afff(u)dxdx Using theresult fortransformation ofseries fa \eflu)=1)Dax”|=DA, (a...Je”Ans J a0 where theA.are functions ofa...) rather than Up,...us. Now calculating theintegral Jffayax ax or SfDAvtdxax=> AstfosHin+2)=> Ax!fio1) Movie DecoMrosimion 145 Therefore Faxtecy+xq,-05, Avant/nin=0 Equating coefficients, a=c, asc and forn >2 a,=-aA,,/n(n—1) Tomatch theboundary conditions, weapply decomposition tothe integration constants and double decomposition tothecoefficients ofthe Maclaurin series solution, => Q=d Ea Ea Substituting into therecurrence relations —_— oer) a - and forn >2 a=-a AS/n(n—1) We have achieved adecomposition oftherecurrence relations which will determine thesolution oncethecomponents c{)andc{*)arefound. Next, weorganize thesolution into aform suitable formatching atthe boundaries byrearranging thedouble decomposition components ofthe solution into astaggered series. This organizes thesolution intotheboundary- value formatsothatc{*)andc{*)components arecalculable byuseofthe approximants totheboundary conditions Gan{u}(x=6)=b, Gani{u(x= &)=d; Ma Cuarren 5 ‘We have Tostagger theseries, uy=a+ax uaa saxax? eax? uy=al?+axsallxteal?eax!+ax! ug=aff)+a)x...all x0)4a) x2 which we can write tes , vga areal Fed where [n/2] isaninteger greater thann/2(forsecond-order equations). For third order. wewrite [n/3} andforfourth order, [n/4]. Wenow have a different decomposition ofuwhichissuitable forboundary-value problems: =SOsetesa) i>ee>iae aa a Now wederive thesolution approximants forthestaggered approximants °Fale)anexal)extal”),Fool Hence C2-:{U}= b501{40}+Xe01{8:}+%°0.{a,} 77°,{as} 70{az} 779, {res} or Gnufuh=2aeictra) {40} Nowusingtheboundary condition approximants > Mooirieo Deconrosmon las ee Gnui{ul(x= 6) =b, Gani{ul(x=,)=b. wecancompute theconstants '*),e{*?anda”,al”thecoefficients for theMaclaurin series determining thesolution. SuMMaRY: The basic steps are: 1)Compute a{*)anda‘*)bymatching thesolution Approximants tothe boundaries. 2)Compute components a,’toimprove theaccuracy ofthesolution approximants using therecurrence relations forn>2.Weget a=3ae a andfinally 7 u=¥ ax" Et 3)Finally wecantranspose fromtheboundary-valuc farmatto theinitial- value format Gani{0s)} =On{20}+banifa,}= with- =Pardo} 4,=bani{ai} andforn>2 a,=-@A,_,/n(n—l). (Note thisisforthecaseofzero input) Thisavoids further needtomatch thesolution (otheboundary conditions. Nowu=") u=>, a,x"andconvergence is accelerated overthatoftheboundary-value formatted solution, LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE (COEFFICIENTS: ‘Weconsider theexample d°u/dx?+p(x)u =0withtheconditions 146 Cnarren$ u(x= é,)=b, u(x=&)=b, Let p(x)= >,p,x* andseekthesolutionintheformu=).™,a,x".Thesolutionis uscy+xe,— ffp(x)udedx Le stays~{Foa'|[Saar]2 it) =z[Saab Now “=fe) [fetedxdx=ff>{Epa dxdx ” aed (ued fe )=YYaabf+1)(n~=2) Ported z =fat )=D[drtbafrie We now have =. =fe lL.Dax’=cotaeD4Deadsnes*/nn-)) Consequently a,=c, anda,=c, andforn>2 a2-Zpa. ss/rin-0 Weconclude withthesolution a{”’=cj")anda\®/=;"!. The ¢,’andc\" aredetermined using theboundary conditions Movin Decourosirton 17 G.1{ul(x=§,)= by Gani{ul(x=§,)= bs andforn2,a”=-S" p,al®)_,/n(n—l)achieving adecomposition oftherecurrence relations. Thencomputing a=)”, a/®',wehavethe solution u,=~, a,x’. HOMOGENEOUS NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS: Consider theexample du/dx? +a(x)f(u) =0withtheconditions u(x=&)=b, u(x=é,)=b; Leta(x)=~, a,x"andseekthesolution intheformu=S", a,x*,i.e. aMaclaurin series. LetNu=a(x)f(u) andwrite Lu+Nu=0 which leads to thesolution usc,+xe,~fff@(x)f(u)dxdx. Tocalculate thisintegral (which isanindefinite integration forevery iteration), wefirst usetheresult for transformation ofseries EF Fd where A,=A,(a,,....a,). Now atsytey={Fax}{Eax}-5 {Seachos Et Ble and ffeGor(u)axax=ffY[Saa,fedxdx =r{Eaa,fe*oeninen 16 Cuarren 5 We now have Faxt=ee25-F{Soro} vo fod Alm Equating coefficients ay=cyanda,=c, andforn>2 Py card Hence, ai)=cf,aj")=c\"?.Using Gaa{ul(x=6)= by Gan{u}(x= 82)=ds wedetermine ci")andc\*)andforn>2 a™=->a,Ais,ota-1) Then.computing a,=~, a\"),wegetu= a,x” HOMOGENEOUS NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS FOR LINEAR AND NONLINEAR TERMS: Consider aspecific example with Dirichlet conditions d’u/dx?+p(x)u+ a(x)f(u)=0 u(x=)=, u(x=&)=b, Weseekasolution inMaclaurin series formu=Y~, u,x°satisfying the boundary conditions. Intheoperator formatofthe decomposition method, the above equation iswritten Lu+Ru+Nu=0with L=d*/dx*, R=p(x)and Nu= a(x) fu). Operating withL*, Moire Decourosion 19 u=o,+x0,~ ffplx)udxdx~ a(x)f(u)dxdx where this isunderstood tobeanindefinite integration forevery iteration, since itis anonlinear function, Following theearlier examples, wecanwrite at ec? ai?) 2c” with c\®’andc\®)tobedetermined bythespecified conditions Gq-1{u}(x= 8) =, Gani{u}(x= 62)=by and forn>2 a=¥path, Ba,as,fon which gives usadecomposition oftherecurrence relation. Then, computing a,=~, al®,wehavethesolutionu=)™, a,x" COMMENTS: Comparison ofinitial-value format andboundary-value format: 1)_Initial-value problem, formatted solution u""Y? LV)A Lv.) ay=Zu 2)Boundary-value problem formatted solution u‘*¥? -> ov)> (m-{0!2}) 50 u=>uf= afer)x! where 299FFeter uh)=SYaleterad xo Ford 150 Charren 5 and where ony ue Weremark further thatu{’”)¢ul®” andg{t¥? #98%), butasnbecome sufficiently large, ¢°”? becomes numerically equal to9°”, ie, u=lim¢) =limg®) Thus theevident difference between thetwo isintheorganization ofthe components ofthedoubledecomposition, i.e.,thestaggering ofthe double series summations makes itpossible tocalculate thematching coefficients. The procedure isquite general andwill work forawide class ofproblems. Icistobeemphasized thattheseries resulting from decomposition isnota Maclaurin series. Itisactually ageneralized Taylor series about afunction rather than about apoint, which reduces inwivial cases tothewell-known series. Despite theimproved applicability oftheMaclaurin series with theuse oftheA,polynomials anddecomposition techniques, thedecomposition series isstill superior inconvergence properties tothemodified decomposition series SOLVING EQUATIONS WITH DIFFICULT NONLINEARITIES: Consider asecond-order (nonlinear) ordinary differential equation inthe form u”=~a(t)I(u)=B(t)withinitialconditionsu(0)=c,andu'(0)=c, Wesuppose ['(u) isobtained bycurve-fitting orleads todifficult computation oftheA,polynomials. Orinafunction such as(uw) =sinuwemay prefer to work with thepowers ofu.We,therefore, write T(u)= Yeu" ‘Using modified decomposition, let v=Dat=ya, w=Dud (nedaa a a mM o=Dan" "=D (n+1)(n+2)a,.20 B=> B Moire Decourosmon 151 T(u)canbewriten J”,A,(up,...u,) andwehaveassumed thatuis written inaconvergent seriesu=J,u,t®sothat T(u)={&a] with u,=a,t’. Hence A,(uy.esest,) =0°A,(d5y0-018,) so T(u)=FA,(29.--124) B Ifwewrite forclarity A,{f(u)} fortheA,representing f(u), Tu)=DALM} =DAs(aoeeet)O ie,,theA,{T(u)} arefunctions ofa,,...,a,. SinceT(u)= 0,y,u*and u'=DF, Aa{ut}™, theA,{u"} arealsofunctions ofa9,...,2—: Substituting, wehave Tew)= Drau”=Drad, As(u)t eo Eire! =Ley 7A.(u") Pore) which wecompare with[(u)= S\>,A,{F(u)}t® sothat AAT} =¥7eAafe"} Retuming tothedifferential equation u”+aF(u) =B.andsubstituting the respective series, wehave 132 Curren $ DY(n+1(n+2)a,,.0° ffae}{3area} =DA Performing theCauchy product {Ear}{Eacroye]-F eSa.astreop sothat Dy(0-902 FS,a.trro}e=d where we can now substitute AAT)}=TA(u’) which makes itezsy tocompute theA,forF(u), Weobtain E(weinsr.0-E1Sa.. Sraueye=¥ ae where theA,=A,(ay,....a,) arefunctions ofthecoefficients fortheseries for u. Equating coefficients oflikepowersoftheindependent variable t, (n+1)(0+2),2+D BoD rA{u") =Be andsolving fora,,.wehave wusBSenSrale!] [o-0-2 with 2= C,and a,=¢,sowecansolve thedifferential equation foru:the A,canbeobtained byarapid computer calculation bydecomposition into simple integral powers ofu,just asthechoice ofLinthedecomposition method ledtosimple integrations, avoiding difficult Green's functions. MoviFteo Decompestrion Iss REFERENCES 1.G.Adomian and R.Rach, Modified Decomposition Solution ofNonlinear PartialDifferential Equations,Appl.Math,Lett.§.(29-30)(1992), 2.G,Adomian, R.Rach, and R.Meyers, AModified Decomposition. Comput. Math Applic., 23.(17-23) (1992). 3.G. Adomian and R.Rach, Nonlinear Transformation ofSeries, Appl Math, Lett, 4 (69-71)(1991), CHAPTER 6 APPLICATIONS OF MODIFIED DECOMPOSITION Wenow consider some applications ofmodified decomposition. The Duffing equation isaninteresting example ofanordinarydifferential equation; ithasimportant applications further discussed inChapters 11and 12.The equation isgiven as u"+au’+Bu+yu'=g(t) CONSTANT COEFFICIENT CASE: We assume that the solution, aswell asthe excitation, isinMaclaurin series form. Then. id= >g.t® ueSale u’=du/dte=S) (m+1ag, eePuld? =L(m=1(m-2)a, 2" fe Ve Je. jos Aaa =D acter ty wO)=k, =ap= Yat hoo w(0)=k,=a,= Y(mslay. ho Subsutuung APrucarions oF MooirieD Decoupostrion 155 F(msifm=2)a,..t2 aS(m+Yaga as mS BY,gt7S,Aalayendet=5gat® Thenos a s (m+1)(m=2)a,.;+(m+ lag, +Ba,+7A,= fq We can now write the recursion relations a,=u(0) a,=u(0) a,=$a7am tay.~Bas~7Ag ae (m+1)(m+2) sothe a,are determined (dependent onthe Aq) and we can write u(t)=7,agt™.Thusuy=a),uy=aytn. ay=u(0) a,=u'(0) a,=(go(1a a,~Bag—7Ao)/(1)(2) a=(g,-(2)e@ a,-Ba,-7A,)/(2)(3) a.=(2:~G)@a,—Ba,7A,/3)(4) as=(g, -(4)a a,~Ba,-7A,)/(4)(5) a,=(2,—(S)a as~Ba,-7A,)/(5)(6) a,=(25—(6)a a,-Bas~7As)/(6)(7) ay=(26—(7)a,~Bag—7Ag)(7)(8) ay=(2,~(8)a,-Ba, 7A,)/(8)(9) Aq=(5—(9)ay~Bay—7Ay)/(9)(10) ‘Theapproximants tothesolution willbegiven by 156 Cuarren 6 0,=a =a) tat Q=a,tajteae Gen=d at For convenience ofthereader welistthe Adomian polynomials forthe nonlinearity intheDuffing equation: Ay=a} A,=3a}a, A,=3a}a,+3a}ay343a Ay= a)+3a5 a,+6aa,a, A,=Sapa, +3a?a,+343ay+6aya,ay Ac=3a§a+3aja,+3ai a,+6apa.a, +698,25 A,=a)+3aj a,+3a)a,+3ai a,+6a,a,a, +68, +6a,.25 A,=Baja, +3a} a,+3a}a, +3aj a,-6a,aay igs +6ayase, +62,a,a, Ay=3aa,+3a;a,+Baia, +3a}a,-Baja, big, +6adsa,+Oates—62,8.85+64,ay2, A,=a}+3aj ae+3aj a,+3a}as+3a)a,+6a,a,a, +6iigt8,+Gigiiyal,+6pesa+68,52, +6ayiya,+6888, Aj)=345dig+3aa,+3a}a,+Bafa,+Sada,+3aiay +6igt,a, +Oagtna, +6iy2,a, +6aga,a, +6a;i,8, +Oaaya, +62,A,8, ~622,85 ‘Wecansubstitute theexpressions fortheA,intotheexpressions forthea, butitisunnecessary, since itismore practical tonumerically evaluate theA, beforchand andthen todetermine thea,.Ifoneprefers towork with thefinal expressions forthea,theyare: APPUcATIONS oFMODIFIED DecoMposirion 137 ay=u(0) a,=u'(0) a,=(8)~(1a, ~Bay~Y85)/(1)(2) a,=(8,~a(2)e, ~Ba, ~(35,)}/(2)(3) a,=(g,~0(3)a, ~Ba,~y(3a9a,+3a;a9))/(3)(4) ag=(8,~0(4)a,~Bas~7a!+3}ay+6a,0,24)/(4)(5) a,=(g,-0(S)as -Ba,-yGada, +3a}a, +3aay+6a,.2,))/(5)(6) a,=(25~06), —Bas~7(3a5 a,+3aja,+3aza, +6a,4,8,+63245))/(6)(7) a,=(g,—0(7)a, ~Ba,-1a} +3aza, +3a7a,+3aza, +a,aa,+625apa+6a,2,2,))/(7)(8) a,=(g,-0(8)a5 —Ba, —7Baza, +37a,+3aza,+Sala, +6ay4,84+6a,2:05+6g2a,+6a,2.2,))/(8)(9) ayy=(84-(9a, —Bay—7(3a5 a,+3aja,+3aza, +3a5a,+3aia, +6a,4,2,+64,0,8,+bay4,8,+6a,a,a5+6a,2,2,))/(9)(10) VARIABLE COEFFICIENT CASE OFTHE DUFFING EQUATION: Write wfZavehre[Sae$ po{Eeratefor= Sese= me mS les Fo) a w=Y(m+tja,,t? a u”=SY(m+l)(m+2)a,,.t™= Es ote (0)=ky=a=Ya,thao a w= k,=a,=D(m+ Daghao 158 Charron 6 Llm+(m+2)a, a"-[E-.eS(mDrea] Ex Ex =} Carrying outtheindicated Cauchy products, Lm+y(m+2)a,.0 +{FeastDiee+>{Saale Fa} am(38 mle J +ESrcalenvalhin=Zee an mn Consequently, (m=1m+2)ay..= Yay..(0+Day, Dect +EeAeltorts)=Be 0recursion relations canbegiven a,=u(0) a,=u'(0) 80~$5{ete-(0*8401+BonettFewA} Be5 im=1(m=2) sothatu(t)= 7,a,t®isdetermined since theA,areknown. The computation iseasily programmable, Wecanlistthea,asfollows: Aprucsros ofMooirieD Decourosirion 159 a,=u(0) a,=u(0) a,=(8,~&5(1)8, ~ByAy~7Ao)/(1)(2) 8,=(2)~G4(I)a~&%(2)ag~ByBo~By;7)Ay~YoA1)/(2)(3) a,=(8: —@,(Ia, ~0(2)4, ~(3),~Byay~B,a,~By; W128 ~TA,7ADIB)(4) a,=(g,—@4(I)a,~oF(2)a,~&%(3)~0(4)e, Bya,~B,a,~B,8~Bya,~75Ay~72Ay~ 7,Az=ZoAy)/(4)(5) a,=(2.—(ay ~0(2)a ~0(3)a, ~0(4)a, ~0(5)ee Baty~Byay~Bya~Byay~Byay HHAgTyAL72As 1)Ay=TrADMS)(6) a,=(25~O45(1)ay~0r4(2)a,~0%(3),-2(4)a,,~,(5)ay-025(6)ay Bety~Bay~Byty~Ba,~Bra,~Byas W715Xa1Ar~7%Ar12AsAaZoAg)/(6)(7) ay=(85—&(I)a, ~24(2)a, ~44(3)a, ~05(4)a, ~0(5)a5 ~04(6)a, G(T); ~Bydy—Bsa ~Bas~Byay~Byas~Bas~Byay W150 715A~7A ~HAS 12AaTAs ~FoAST)B) a,=(g-~,(I)a,~Of6(2)a,~5(3)a,—(4a,~O(S)as~OF(6)ay —0%,(7)a, ~O9(8)a,-Byay—Bya,-Bea:-B.a,-Ba, Bz5~By8g~Byty~14Ag15Ar~5Aa~TesTrAe 12As—NAs~ ToAr)IB)(9) yp=(83—(Ia, ~0,(2)a, -O,(3)a, ~0(4)a, -o,(5)as —(6a, —&,(7)a, ~%(8)a,~%(9)25—By2,~B,a,~Bea,-Ba, ~Ba,~Bys~Bya5~By2,~Body~YxAg1Ay~7sAa15Ay 1X6 TAS TAs ~1Ad YoAs)/(9)(10) u(t)= 5ay =a, O,= a)+ayt Fd 160 Charen 6 REMARK: Possible areas offurther investigation include regions ofconver- gence,numerical algorithms forcomputation, application ofconvergence ac- celeration ansforms”, andstochastic versions oftheDuffing equation where wesolve forfirst- and second-order statistics ofu.We can, ofcourse, con- sider special cases ofoursolution such as (uO) =k, andv’@)=0 with g=0 (i)u(0)=0andg=Owith u’(0)=k (ii) u(0)=0 andu’(0)=0with g=go,aconstant (iv) uO)=uO) =0,g=gor gytorg=g,+8,t+ Bt" ©)u@=k,wO=k, =D7,gt? ‘Wemight then usedouble decomposition andwrite =D Yea=Coe =D Yas with “ 4,=Uy= Ure 6=6.4u,26,+ 5uyFo Weemphasize thatthese problems arealso solvable byusual decomposition. ‘Also, wenote that therateofconvergence ofthemodified decomposition only approaches that ofdecomposition when theexcitation approaches zero. The reason forthisisthattheinitialtermcontains onlythefirsttermoftheseries forg;only when wegotosufficient terms ofuwillwehaveenough ofthe input tohave asgood anapproximation. “Such asPadé approximants, Shanks andWynn wansforms, andtheEuler andVan Wiingaarden transforms Arpucarions oFMooirieDDEcouPosiriow 181 APPLICATION TO LINEAR PARTIAL DIFFERENTIAL EQUATIONS: Suppose webegin with theequation L,u+L,u=0 and, tobespecific, choose L,=3?/dx* andL,=0*/dy’. Following thedecomposition pro- cedure, wewrite theequation forthexpartial solution u=0,-LI Lu where ©,=4,(y) +x6;(y) must befound from thegiven boundary conditions andL7!isdetined asthetwo-fold (indefinite) integral ff()dxdx.Ithasbeen demonstrated previously thatsolutions areeasily determined bydecomposi- tion. Now, however, weusethemodified decomposition method which we have discussed forordinary differential equations. Thus welet w= Deas xy onaS u=Die(yR™ where- aaly)= Dee¥ We now have. Foaly)x®=Sly)+8)-[f(2"/2y") Eacty)x®axax a aS Sasi =86)e86)- SE a=Sly)+x6\(¥)~Ya te Dalya? =Sly)+x69)Lier oyel) ~ we = gt a =&(y)+x8(y)- Y—— asDaalve” =Sly)+x69)eres=paytes) ‘The coefficients areidentified by ely)=Soy) a,(y)=8,(y) andform>2bytherecurrence relation 182 Cuarren 6 ag(y)=-2/dy? a,.a(y)/m(m -1) ‘Wecanequally well consider theypartial solution bywriting u=0,-L} Lu where ,=no(x)+yn(x) andu=7, D7,caeBY u=Sb,(x) 9” Eat where b,(x)= S7,CaX®andLj!=ifff(dydyisanindefinite integration operator. Now Db,(x}y" =n(x)yn,(x)-Sfielex ydb,(x)y"dydy =n(x)+yn@), —— #9,(x) . &(n-)(ns2)ax * = yet gt=n(x)YIOEMas)ae) Wegetimmediately bo(x)= M(x) b,(x)= n(x) and forn>2.balx)=(-0°/2x" jb,_-t)/n(n1) We now consider themore general linear form Lyv+L,u=Ru=0 again with L,=0°/dx* andL,=0°/dy*. Thexpartial solution isgiven by u=0,-L) Lyu-Ly Ru Forsimplicity, wechoose R=p. Armscarons ofMooirien Déconrostrion 163 ©,=Sly) +xG(y) uedYay a=¥ay) where a,(y)= D",Cas¥*andLi!=ff())dxdx isanindefinite integration operator. Now Ea.)x" =80) +80) [ead a(y)x®dxdx F-} ~ffoX, ag(y)oxx Eao)" =80)+ 80) & = yt gt 2wep ag) en ~Learae +2)#a(y) ¥a(x=&0)+x40) ~ 2 3xmma) dyte) Day ae2(¥) sothat a9(y)=Soy) a(y)=S(y) and form >2 164 Cuarrer 6 (yeFLAY Janal¥)= Peal¥)a ‘Theypartial solution follows similarly: bo(x)=Mo) b,(x)= m(x) and forn >2 ; b)=(-F/By")bq.2(x)—pb,-2(X)n(n-1) Weconsider L,u+L,u+Ru=0. LetR=p(x,y) with L,andL, asbefore, PXY)= DD Pak” p(xy)= >Pely)x® where - Pal¥)= >Pas¥” u=,-Ly L,u-L} Ru ®,=Sy)+x8(y) u=Ya, (y)x* where: = ag(y)= >Cue¥" Now“ Ess(v)s"-80)+50) eary, agly)x®dxdx “IE,extore}(E soyahaven Werewrite thebracketed quantities fe = le.s{Eoat}Esso"|=¥«°Flon.) Now xa(y)x® =S(y)+xG(y) 2(mrhmsy ay=) >»jrnmsy PalY)Ba-u(Y) ¥sab)s"=80)+580) = x ae-.>mma ay220) sothat ay(y)=So(y) a(y)=,(y) and form>2 (-27/8y")an-a(¥)— YPul¥)@e-2-4(¥) wa(m—1) Theypartial solution follows similarly 165 Charren 6 bo(x)= mo(x) bi(x)= n(x) and forn 22 a (-818x')o.2(0)~ YPolbs-2-08) b.(x)= Pda(x) ane) We have used R=p(sy)= DY Park y=>palxdy® an 8 m where Pal)=FPne®” Also,v=7b,(x)y” withb,(x)= 7ca.8”and unDD aed mis APPLICATION TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS: Consider L,u~L,u~Nu=0 andthexpartial solution with L,=d*/dx*. L,=F/dy*.andLy!=f{()dxdx, WeletNu=arf(u).(Wehaveshown algorithms forconsidering functions such asf(z") orf(u.v,w) inChapter 3.) We have now u=,-L;'Lju-L;Nu where ,=Ely)+x6,(y). Let vad Yocgaxtytus ¥agly)x® am Eo or a,(y)= xCon¥ Wehavegenerally written f(u)= 0”,A,;however, weshowed inthe Appucarins oFMooiiep DEcourosmiow 67 previous resultsonthetransformation ofseries thatifu="a, x” wecan then write Fu)= Dx Ay) where A,,(y) =A,.(29(¥),.-2_(y)) andwedosonow, Wenowhave ¥aa(y)s* =0)+80) -[(wayS ag(y)x®dxdx 7) -faxx"A,(y)dxdx Ea.) -S0)+80) = et gt 2any ay ae Zero) >ag(y)x™=Sol¥)+E:(¥) ee B ~ oe Damn Se) sothat oly) =Say) a(y)=4(¥) and form>2 _(21Ay*Jago(¥)~ &Awaly) aa(y)-—aa-) Theypartial solution issimilarly obtained, letting u=.~,b,(x)y® where bi(x)= Dong Ca¥ Oru=D.Don gCanaX*y*Let 108 Charren 6 f(u)= Dy BA) withB,(x)= B,(b9(x),...,.b,(x)) wherewehavenowusedB,instead ofthe usual A,only todistinguish itfrom theprevious setofpolynomials. GENERAL INHOMOGENEOUS PARTIAL DIFFERENTIAL EQUATIONS: Considerthelinearcase:Lu+Lju+Ru=g u=Daa(ye* ag(y)= Sly) aly)=Sy) and form>2 Fac) (0°193°) ay.(9)~Lo(9)te-s-0(9) a,(¥)=—$—$$&as ——————__tr am(m ~1) where R=plxy)= Dpaly)x Pald)=¥ Paod® 2=a(sy)= D7e(y)x® Yal)=¥ Sa0¥" ©,=Bly)+x5(y) Now thenonlinear case: Lyu+L,u+Ru+Nu=¢ u=Da,(ye" Nu=a(x,y) flv) aly)=Eo(¥) a(y)= 6(y) Arpucsrinsof MupiFt@D DecoMrostiow 169 and form>2 ata,(y)={rantd)~(#/2Y* Jags(Y)-Dea(Y)aa2-a(Y) Sa,(etn HN) where“ Aaly)= Aa(ao(y).-Ba(¥)) and @=a(x,y)= Da(y)x® ‘The solutionis- etine aS ay,=02 a,=o! =VagGig,~B(0+1)(0+ 2)5-2 fart= (m+1)(m+2) Consider theexample +Qu+pB-7(tx) mea" BS Ugg=ayatx" Yee a=zYim+1)(m+2)a,,,,0% ==>dea+1)(n+2)a,.,2t7X” 170 Cnarren 6 u(0,x)=6(x)=}6x" Boxe) ¥ex?a Ford The solution is a,=6, aie =F ao ZanMe ~Bln+1)+2)geen‘woke (m+1)(m +2) ‘The algorithm hasaduality property since wecancalculate either thetpartial solution aswehave orthexpartial solution. Ler’s take B=1towrite aw YaaWes(041/02) a (m+1)(m+2) forthexpartial solution. Ifwehave aboundary-value problem, weusethe double decomposition technique forafewterms togetagood initial value. ‘Then rearrange andsolve asaninitial-value problem. ‘Summarizing, thet-coordinate partial solution—we might callitthetemporal format—is ved|Sacax|e=3aon" as = with theuthapproximant S/F. ele 2=D |Dae i©aa(x)t Eat a while thespatial format ofthesolution (orx-coordinate partia! solution) is w/e) « v=> |Dat” =>a(x" ost \m=0 / bao APPucaTions OF MopinED DECOMPOSITION i with thevthapproximant a=d(Sane}e-Yaor’ ae? (ano aed Each sequence ofcoefficients inthetwo formats, i.e., thea,(x) and thea,(t) hasitsown radius ofconvergence, and moreover, thesolution itselfisunique. Asanother example ofalinear partial differential equation. consider Fu, au du_.Fu_ SS. anStat +Bury -5l5= Fae Poaa Pap> antX u(0.x)= >)pax*a 2u(0.x) _ox @.B,7.6,€ constanta w=DD tax” BS Bn. =Pe ay =O, a,=faeO(M +Dagng—Baas~1(0+1)ap001(+10 *2)89.0-2 are(m+1)(m+2) Wecannowwritetheapproximants ¢,.=0")Dir,a0". LINEAR PARTIAL DIFFERENTIAL EQUATION WITH VARIABLE COEFFICIENTS: Fuf[SS, wld [SSy, wolyleSywelsu aoe ax pees Jat°x*bu TagOx? FFseePte Fee 4H bast =DDeaalx anoeo Ox mae u(0,x)= px" alos)“Sex 172 Cnarren 6 Using modified decomposition wewrite uaDD aeths* LINEAR PARTIAL DIFFERENTIAL EQUATION INTWO SPATIAL DIMENSIONS AND ONE TEMPORAL DIMENSION: Fu du du, .du_,Fu Fueas Busy pe eheal An RRRRRLNe HDL DYFeaat’x*y” ul0.xy)= 2DPaar" Se a SF gameFvl0.ny= DLoeax*y The solution bymodified decomposition is Bam Bree=Paw Bee =Fme Bysren={EemeAEDByi BA.~MFM)Beae ~B(0= Dyan ~E(m+1)(m+2)a,0.25 Han =1)(0+2)e.ne2}/{(C> IMC2)} LINEAR PARTIAL DIFFERENTIAL EQUATION INTHREE SPATIAL DIMENSIONS AND ONE TEMPORAL DIMENSION: ca ou du ,du_ duLYGBBysys52hee Be ayPEtaOTeS, a sme nob+pPheX- eaumeOXY Arrticamons oFMooireDécourosmon 175 ul0.xy.2)= 5YSprwax'y2" a ees tymeFoxy2)= TLSFaax'¥2 Solution bymodified decomposition: Bytom =Pre Bieme =Frm0 Arena ={Excme HK*DRce Ba cae H+ DAcree SUM +N cane H+1%goons“MEE+2)pars ~$(m+1)(M+2)2parr ~0(0+10 +2)emara}/{(ke+ IKK+2)} NONLINEAR PARTIAL DIFFERENTIAL EQUATION: aud u ou au .au 2+et myroinATL Paa-zxEqgt™x u(0,x) =YA," 20x)_S5yeend oF Assuming constant coefficients, write w= Fayex > wheretheAj.areourpolynomials. 174 Charrex 6 Aon =Po ai =Oy Qee2a={Exe(M+1)dp..4~Bdge710+1)de2.y ~5(n+1)(0+ 2)aq.2—VAna}/(m +1)(m +2) ‘The approximants will be ae= DDaaot'x® NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS: Fef[SSeells Speelaele Fne a Jax le } HL Deets? u(0,x)= px? 5 « Zyox=¥ ox"Fux) 2o,x Forthesolution bymodified decomposition. wewrite wD Daeet we DDAnots® wherethe4...areourpolynomials. Bon =Pr Arvucannns oF Hoownn Secinosro" Vs [| ae Bana =)Ema7DD anal *Daye tC" SS HEY Byron DYPocusral?* D2 EY SasaelVAD042 Bon -2Even] foY(m+2)} APPLICATION TO COUPLED PARTIAL DIFFERENTIAL EQUATIONS: Consider thecoupled partial differential equations Fu eu wee =B(x.t) av avSatSetrus(x) with (uncoupled) boundary conditions™ u(x,t) =8(0) u(x,0)= (x) du(x,0) uest)= (9) 229) 509) t wrat)= (0 ¥(%.0)=6((3) Av(x,0) voxnt)=m0) 269) Ga) Wewillusethespatial format, hence theinitial conditions arenotused inthis example. Weassume oand7areconstants and Blxt)= YD Bas I= >Fare **Notethatthecaseofcoupled boundary equations issolvable. 176 Charron 6 gM=y He Ex} eW=y oe no=y me n()=>Pe Define 2Lo=530 L()= A(t)+xB(t)+ ffdxdx fortheusolutionandC(t)+D(t)+ [[()dxdx forthevsolution, (Wepoint outalso that wecan usedouble decomposition and recast this asaninitial- value ortemporal format problem toaccelerate convergence.) Webegin with Lu=B(x.1)-(#/aF)u-av Lv=8(x,1)-(2'/avv— yu or usuy-i (eer u-L? av vey Lb(8/orw-L rv with Ug=Ag(t)=xB,(1) +ffB(x.t)dxdx Vo=Co(t)+xD(t)+ ff5(x.1)dxdx Nowwecanwritethedecompositions u=7,u,andv=, v,andthe approximants o,n{u}=> uy, Et Av}= 2 Avruicinons oF MeoireD DEcouposiTion a7 ‘The general decomposition components u,and v,are: u,=A,(1)+xB,(1)~ ff(F/9t?)u,., dxdx —ffovendxdx vs=C,(t)+xD,(t)= ff(#/a)v,., dxdx Shyu dxdx We have. ofcourse. 9{u}=u, ov} =. a a dulu}=> wy Gufvh=d ve ‘Theapproximants must satisfy theboundary conditions; hence a,{u(xt)} =800 9,{u(xa0t)} =E300) o{v(x,t)} =(0) 4,{v(x20t)}=(0) which implies Uo(Xist)=S(0)Uo(x2+t) =&(t) valXst)=m(0)¥o(Xast) =M(t) Also o{u(st)} =(0) Safa t}}=G(0 os{u(st}} =&(0) Grnf{uleat)}= &(0 d(x. h=m(0 Aia{Md}=m(0 1{v(x2.t)} =m(0) in{r(xnt)}= m0) Since 4..Au)= o,{u} +0, Oatvh= Oty} ty, weclearly have tg(Xt)=ua(2,1)=0 vast)=v_(x55t)=0 Summarizing, u(t) =&(t) 42.1)=E2(1) volXst}=m1(0), vo(X3.1)= a(t) andform31,ty(Rist)=ty(Rast)=ve(Rot)=Vo(ast)=0.Write Ag(t)+XBol(t) =6:0) Aa(t)+x2By(t)=2(0) Colt)=xDe(1)=1:(2) Cot)=Dot)=m7) and Adit) HAC) =FC) A(t) +x:B (Y= 50) Cal eDA(t)=mf"(0) C(t)+xD/(=(0) Now E11)=E(0-[JBlx,.0)dx, dx, 8)=£.(0)~[JB(x,.-1)6x,dx; nt)=n(t)~Jf6(x,.1)dx,6x, n'(t)= n,(0)~Jf(x2-t)dx;de and EM)=[f(F/2e uy.(x.t)dx,dx, =ffevea(x.t)dx, dx, EM)=ff(FAvVu,,(x..t)dx, dx, affev, (xs.tjdx, dx, ApricaTions oF Mooittp DEcoMPosmiow I nit)=ff/A2Wv,. (Xit)ax,dx, +fJrecilxet)dx, de, TH(0)=[f(/007)ve (Kae)daydy +ffrea(st)dx, dx, Wecanwrite . aM=y ae P=Ene =>,me and“ SHM=y sae Fd a= Be (=¥ge a no=>, mae Consequently, wewrite thematrix equations 1*)me(0) (x)(B00) (*)(60)(0) 1) (D0) (Po, (=.) (200) 1x,) (By) (6°, 180 Curren 6 1x)(C(0)_(n) 1x) (D0) (no, Thus, we find _EK EY, Ag(t)= 2 B,()== SPO ° yo . 3% 1) 300(0)_Sie=SiBee %y—Xy10) 10(=x MCE c=22Wea© b(t)=22=0) °Xp Xp donq=SMe =x)CP)= oqye ame xo 0Ace %)— p28 aX a XX B=— cape SEW xn) “ XX p=Wan) ’ Xy—X Arpucarions oFMODIFIEDDécouposirion 181 coo2Me . Xy—X, po=anne 8 gk Now wewrite theup,V)Components ofuandvandthe@;approximants for uandv uu,=Ag(t)+xB,(1)+ffB(x.t)dxdx Substituting A()=3, age B,(t)= >BOY Blur)= DDBaar? Eceen we have tae Bate =AB, 7 att =o, uy=Ag(t)+xB,(t)+x>>jarh(msd o,{u} Similarly, Vo=Co(t)+xD,(t)+ff5(x,0)dxdx with cy=y coe D()=y Pe Et (x)= YYSaxe sothat “se Vp=Co(t)+xD,(t) +x?¥xyboaUgty. .a&(m+1)(m+2) " We can now write Us,voinaconvenient form as 182 Courren 6 wed Fare wed Dba me where anAn ay= BY? a),=Pes ste (m+1)(m+2) and bil=c® b= Do = (m+1)(m+2) Next, wecalculate u,andv,forthe@,{u} ando,{v} approximants u,=A(t)=xB,(1)—ff(2/9)uydxdx ~[fevedx ax Vp=Cy(t)+xD,(1)= [f(2*/2E vodxdx —[Jrecax dx Let Aged abet B=) Be qed cr Dwy=d Dye Apeurcarions ofMooitieDDécourosmiow 183 _ Hee FLANM+2) 40)pepe u,=A,(1)+xB,(1)-x’=zz(mei(maaj tre 25 FH be ee“2D>(mzh(m+a)** = Hey F(92142) 0 pope v,=C,(t)+xD,(t)-x’ z>imzim=2) beh,x" 1 F Yass yaa“2S Garni We can now write y= YW ee wed Fote where “ee ah=At? ai=By) a,=O D(a+2)a8.,=abe ond(m+1)(m+2) and bea=Cy? bia=D.) po,=FD+2)b2.n 7a ere(m+1)(m+2) Now wehave theapproximants 9,{u}and9,{v} o.{u}= {up+u, orv}=olv}+y, Now wewrite thegeneral components u,andv,and theapproximants Geoi{u} andO1{} u,=A,(t)+xB,(t)=ff(#/t?)u,., dxdx-ffav,,dx dx ve=Ci(t)+xDA(0)~ff(F/2U verdedx~[f7u.dx dx 186 curr 6 age, ave B=Eae CA)=5cle b= vite Substituting = ws Dace wehave weAft)exe (eS FOLNO*2 goryee(O-8XE Gmeymen ee ee ee eT“22 Gena eeCilt)aden SFAANA2) yenme xD/(t)“Zz im+iims3) bys, x oeSS te cee“22 Gey andfinally uaXxaxett ved Fvxe where “es aio,=FM(n+2)aes =aE ese (m=1im+2) Arpucarions oF MooiFieD Decomrosirioy ras bach, bl=DY pio,=e Din+2)b32, =7a? aoe (m+1)(m+2) Now theapproximants 4,,,{u} =@,{u}+u, and9,.,{v}=0,{v}+v,. Tiwewrite@,..{u}= D4,u,and¢,.,{v} =y.,vy,andsubstitute wed Fee we can write .. aotv}= DYYaar =DDYGrftea bere ean da}= dyDYbare ad ~ =e-EESuhre am lS =DYDbuifdon}eret and we note that fimesatu}= DDfimOra{daahee™ or =D Dare andanalogously v=>Yd a ot 186 Charren 6 Tosummarize, u=7,u,andv= 7,v, w= Dal xe wad Yboet ued Saere vey Doe at ane a ban=Ddi COUPLED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS: 28FO,uveBx.) ax ot av ay LY yuv= a(x,Bxae7TUYeCR) with uncoupled boundary conditions"** ulxit)= S(t) u(x,t)=E(t) v(xyt)= mt) v(x:.t)= M(t) Inoperator form Lu= B(x.t)~(0°/d1* ju-auv Lv=8(x,1)~(0°/at jv-yuv ***Wenotethatthecaseofcoupled houndary conditions isalsosolvable bythedecomposition method. Arpucarions oFMoviFieDDécoMrosiiow 187 Operating with theinverse operator (anindefinite integral operator) usu, -L'(e/dt)u-Liaw vevyL'a /at)v-Loyuv where uu,=Ag(t)+xB,(t)+ffA(x.:)dxax Vo=Cy(t)+xDo(t)+ff6(s,t)dxdx Let u=¥ u, ved. Es Ss w=iEo} {E+}-£ Svar This nonlinearity canbeexpressed interms oftheA,polynomials . A= tay where “Ay=A,(UpseoenttpiVoree¥e) Wehavechanged thenotation ofthepolynomials slightly, i.e.toA,,sothat itwillnotbeconfused with theintegration constant A,(t). Theapproximants are$,,,{u}=F, uyando,,,{v}= Dy,veandthe components u,and v,aregiven by: u,=A,(t)+xB,(0)~ ff(/ar?)u,., dxdx et-ffofSoivaisFad v=C(t)+xD,(t)=ff(3"/2t?)v_., dxdx -ff15Unavipa & Since 188 Curren 6 =DY alyeet on 8 theterm u,.,.;,¥, canbewrittenas [$Earee}{E See Deere yy alee BR Me which issubstiruted intotheequations fortheu,and v,components, EXERCISE: Generalize thealgorithm forthe A,polynomials toproducts suchasuvsobserving thatu=x,u,and wed Yuu V=SY Yessy, Aw]=d Yu uu, EXERCISES: 1) Show thesolution oftheanharmonic oscillator u"+au'=B()= >,Bt with u(0)=¢,, u'(0)=6, isgiven by ‘AppuicaTions oFMooitteDDECOMPOSITION 189 with af)=cy, at”=c ath,=Ba/(n +In+2) form=0andform>0by gio) =___~@AS**(eme(n+ms2) where AP)=P" Ye,aera. 2)Generalize thealgorithm fortheA,polynomials toproducts such asuv observing thatu="~, u,and WDD teat mo PADDY vest ats oS it a . ra Adv]=2 2seats Rim CHAPTER 7 DECOMPOSITION SOLUTIONS FOR NEUMANN BOUNDARY CONDITIONS Forsimplicity, consider alinear differential equation Lu+Ru=gwhere L=d°/dx* (and Rcaninvolve nodifferentiations higher than first-order). Assume conditions aregiven as du/dx|,.4, =B ula|sas,=Bs Thedecomposition solution isu= +L”’g-L"'Ru where @satisfies Lo =0andL”isapure two-fold imtegration (not involving constants). The derivative du/dx oru’isgiven by u’=0'+Ig-IR’ whereL@’=0andIisasinglepureimegration andis,ofcourse,notequalto thetwo-fold integration operator L’',Returning nowtothesolution u,wehave bydecomposition: u=+L's-L'Ru Y= Lo. +Le-E RYvy, where wenote thedecomposition notonly ofubut also [email protected] decompositions arealso sometimes useful. For example, when integrations such asL“Ru orofLg become difficult, Rorgcanbedecomposed into 4convenient series sothat theindividual terms become simpler integrations. Now u,=, +L''g and 2B > = uy=,-LRu, =¥(-LR)'®,_, +(-L7R) Lg sothat Decowrosmow SouuTionsFoRNevsaNBOUNDARYCovorTions 191 2 u=SUy (-b'rs'o,., +(-L°R)"L's 5 =Scere oer)" 'e u=¥ (-L'R)*{0+L"g} Differentiating uandnotingthatRucanbewrittenRiu’,wehave v=’+Ig-IRW’ Wenow solve the u’equation bydecomposition just aswedidforthe u equation: Yw=dop+e-RY wy Bs a Hence, Yu=y Oo+le-RYwymS a u,=05+1g and for m> 1, uy=®%-IRIu,., =DCryo, +(-IRIPIg sothat .. ~ w=SYD CRD... +(e faes =D{Cry* Yo+(-mRy"Ie} Ea Ed or. v= (-IRN*{0’+Ig} Eat ‘Wenow need todetermine’ todetermine theconstants ofintegration c,and ¢;involved in©’orcoqand¢,involved in®,,.. Forsimplicity andclarity weletg=0andcalculate theseries foru.Beginning with Lu+Ru= 0,we have 192 . Churren 7 u=0-L'Ru @=c, 40x a=P(E RYO=D (R)"(c +014) =(c5+¢;x)-L“R(c, +¢,x)+(L"R)(cy+¢,x)-... The series for du/dx is . =i = -py, “py, w=(dldx{oy#¢x-L RcLRex+(L"R)¢,+(L"R)ox--} =¢,-IRe, -Rex+IRPRe, +IRPRox.. . Noting that Ie,=xe, vu’=c,-IRIc,~IRic,+IRVRe, +IRVRi¢, ~... Rearranging andcollecting terms, wehave u’=(c,~TRe,)=IRI(e,~Teg)+(IRI)"(¢,~Rey)=.. w=0’IRI’ +(IRI)@’=u,-uptuys... Thus ©’=c,~IRey =, ¥IRe,,. andwenots that (d/dx)u, #Uzbut DY(wexju, =Fuy i.e., thedecomposition isnotunique. Thus, although dug/dx isnotthesame asthecorresponding derivative ofthemthcomponent ofu,theinfinite sums are the same. Returning tothecomputation ofthesolution ingeneral andmatching the solution tothegiven conditions, Decowrosiriow SouimowsFoRNevMawBouNoARYCONDITIONS 193 Uy=Co+XCyo+L" 1=Cy~Rey, +18 =u, 9i(b)=B, 94(b.)=Bs which determines Copand¢9,.For g=0,thematrix equation determining the integration constants is” ~{R(b,ab,(eu)(8) ~JR(b,)ad, 1)(eo(a, Uy=®,-L' Ru, Oy=CogFXOg uy=) -IRIuy, : 2e%,=> Matching 9%,,, totheconditions, Gani(b,)=B, 24(b2)=B: determines Cy, and C,.,. Thus ~JR(b,)ao, 1(::)-Boa ~JR(b,)ab, 1)(ee) Bro "IER isconstant, forinstance R=p,theequationforcyandCyyis “Pb 1)(ee) (8, pr, UW le) 198 Cuarren 7 where §.. andf+, aredetermined from Fan =U +%% where@%.,represents thesumJ’,u,. Nowtheconstants C..n,Chmare determined forallm.Henceallthe%,and©,aredetermined. weed CLR) OL +CE Re =re= =a] v=SEery on.een) bg] Upon rearranging terms STR) F =p) Lg] v=Severs Eosree]Eo Fx} 3 u=>(-L'R)"{o+L"e} where d=S",o, Wehave shown atechnique forlinear operator equations fordevelopment of aninvertible matrix forthevector equation, which determines allconstants of integration. This method isreadily extended tothenonlinear case andalso 10 Partial differensis! 2quations. Consider anexample: letR=|andg=0.(Then IRI= I?=L",) Wehave @u/dx? +u=0. Substituting g=0andR=1inthepreviously derived solution u.wehave veFce'wy [email protected] eg un5(1R/(o+L's} Wecompute u=c,cosx+¢,sinx u’=—c, sinx+¢, cosx Décomposimow SoLuTIonsrokNEUMANYBoUnoakYConorTioNs 195 Matching u’attheboundaries du Zi(b= Bslb =B du Xy,)=B:galt) =B: Then =c4sinb,+c,cosb,=B, =c,sinb,+c,cosb;=By (7sb,cosb,)(co)_(B,)\-sinb, cosb,)(c,) \B,) which. foranon-zero determinant, determines coand c,andaunique solution u=C, cosxc, sinxsatisfying thegiven Neumann conditions. SUMMARY: We have shown thesolution forNeumann conditions oflinear ordinary differential equations. Theprocedure canbesimplified considerably forlinear differential equations butisageneral procedure fornonlinear differential and Partial differential equations forboundary-value problems. Theprocedure of decomposition oftheinitial term, aswell asofthesolution, yields faster convergence because when wehave found ann-term approximant, the resulting composite initial term wpincorporates more ofthesolution. and hence weaccelerate convergence. CHAPTER & INTEGRAL BOUNDARY CONDITIONS, We first consider anexpository linearexample: duldx?+yu=0 with conditions given as: f: uex=§)=b, +f"8,u(adex u(x=&)= b:+fB,u(x)dx y-Bj.andB,areassumed constants here although they canbefunctions ofx with minor modifications totheprocedure given. Indecomposition format we have Lu+Ru= Oor L*Lu= -L'Ru or where [}isatwo-fold pureintegration withrespect tox. Since c,—¢,xisidentified asu,,wehave u,=—yI2u,, u,=-yIiu,,.. Consequently, wewrite =D (7)legx(2v)+¢} "(20+ 1) Since thesuccessive approximants @,represent u(oincreasing accuracy asm increases, each g,must satisfy theboundary conditions form=1,2, When m=1, (G,)= Uo(E,)= db; GCE.)=Uo(Es)=dy or C9+0,6, =b, cy+¢,8,=b; thusthe“matchine coefficients” ¢.and¢mustsatisfy Irecras Bounoant ConotTions 17 fi§](el-(r] UeSiles Lbs The and aredistinct points, i.e., &#6,, inatwo-point boundary problem; hence &=(xb, ~4b:)ME -§) ¢,=(b,bE -4) Next wedetermine 9,=9,+u,. Since @,(2,)=b, and9,(é,)=b, already, and ; E)eb,+[*ox(8)=b, +fFBoCoax (Ge)=b.+fBro(arax Equivalently, uG)=£Byto(x)dx 4G)=fFAruaCdax Thenextdecomposition component isu,(x) =~Y{¢y x7/2+¢,x°/3!] andwe form 9,=9,+u,which wematch tothegiven conditions &+E,x-71ex?/2+q,x°/3!]=d, +ffBe+enax orvt fo46,X=Bet[AGHaetlx/2+Gx°/31 +}, B Therefore tomatch theboundary conditions, be Gegg,-fBID+xydx+oPE1240 2131] +E =EBull?+xydx+712/2402/31] on We have added thesuperscripts (1)todistinguish new values from the previously calculated values ofcyand c,ontheright-hand side. The right- handsidesofthetwoequations aresymbolized asb(”andb{”.Then 198 Cuarren 8 +e &<b? ee &=0 which issolvable asbefore foranew c,andc,. Asmincreases, @y approaches theuvery closely sotheerror vanishes. AMORE GENERAL PROCEDURE: @u/dx?+yu=0 with the conditions UG)=b,+f",wlardx uGG,)=b, +f”B,ude Wewrite Lu=d°u/dx’ andL'=I? where Iisthetwo-fold indefinite integration operator. Then Yu, =o-L'RY u, af where uy=@=c, +¢,x anduz=(-L™R)*@ or uc" y®2(c,ex) sothat = wo ee u=c, -1Py®+0, ¥(-*y* LM oe ZO aaa Even though thesums arerecognizable, write WC Ha(x)+6;HAC) where #4,and11,represent thesums. Next wecan evaluate theconstants ,and ¢,attheboundary conditions. (We call them matching coefficients.) Thus ©Me(E.I*6,HCG)=b,+65Bfsylxrdx+e, Bfwddx CyMe(E:)#6,Hy(G,)=bs+6yB:J"piy(x)dx—c, By”y,(x)dx Inrecrat Bouma Conortons 199 which we can write as G,,Cy+O2.C,=b, 69 +Og, =By bydefining Locate&,,={ut&)~Bfua(ndox} a,~{ucd)-B [mcoas} ésca=(ne)-Afwn(sdax} =(uc)Bfmoar Forming thevectorequation forthematching constants, G%,2) (Cy)_(b,) @%,G2) eq) \bi) Ifthe@matrixisnotsingular (a,Of.~04,@%,#0),thenwecandetermine itsinverse awdnanaeaa8,Oy sothat - t=a'b Consequently Gocy=Seb, —225, °ah ah / a, @, ‘ c,=-S10, +p,lof" Jol* which we can write as cy=biztiabe Gy —O30, c=Sabana by Oy —Ora, Wehave now computed u(x)=Cofy(X)+e,14(x) 200 Cusoren 8 Where ofcourse inthis case, g(x)=cos(4/7x) Qyx (x)=sin:v7 ANOTHER ALTERNATIVE: Start with theexact integral boundary condition: ug)=b,+f"Busdérr Letusdecompose theequation andtheconditions u(x)=D u,(x) andu(G)=Pu,(G,) hence -“ “ XYuG= +f"BYvcoex Then ° u(S,)= dy u.G)=[PBuodx (n>) Now weuseour earlier notation [1]where 4isagrouping parameter for collecting terms. Thus uG)=b, +aeButxrax w=>u,G)2 u(x)=Pu,(x)a Substituting. =w=b+alBS1,(x)2°dx Equating likepowersof2, rece Bounoant Conormions 201 Bs ugG)=b, nat aHS uG)=[Bus(odx ad then weset4=.Since each u,(x) hastwoundetermined coefficients, i.e., thec;"andc{"”,wenotice thatwehave analgorithm tocompute c;f’andc;"” without explicit reference tothem. However, theearlier way ofwriting approximate boundary conditions @,.,,(é,) =b,+iBo,(x)dxisappealing because thelimit asmapproaches infinity isexplicitly theexact boundary condition. EXERCISE: Carry outthecomputations forthematching coefficients and verify solutions. REMARKS: The single decomposition (where wecarry along theconstants of integration) isapplicable toalllinear ordinary differential equations foreither initial conditions orlinear integral boundary conditions. (However, ifthe boundary conditions arenonlinear, weneed tousedouble decomposition. If wetrytouse single decomposition with nonlinear integral boundary conditions, wemust solve forthematching coefficients Cyandc,asroots of algebraic ortranscendental equations, then findwhich roots arecorrect. Itwill bemore simple tousedouble decomposition insuch cases.) Wewillnow use double decompostion forthecase already considered bysingle decomposition. UsING DOUBLE DECOMPOSITION: ~ = \ c=Dem Ec) o=Fe a a=Sul a sothat 202 Curren 8 We have previously shown forboundary-value problems byapplication of staggered summation (see Appendix I)that >y ee Fy)uo = Us Substituting into theequation, we have- - “ = Uy=cy”=xe{) u,=f =xe=7Fu, uy=cb"+xe)" —7Tua or u,=cf"=xe!” 0 ay woXei!& uy=ef!+xef—yey"70 a=ofexcl?pelye SapecZaytci”S =cp 75VettHeyTaye acPaxelpelRenzelBay?gtXEuy=cp)xe—705"meV47?ce ‘ ‘ : et Xs coXs coX. sPeL—poe —poe Peg Tegra uy=Yr) {el x*/(2n)!+of"x"!/(2n+1)}} where recess Bouvoary Conortions 203 u,=due? and ~ ue=(~y)*{of'x**/(2n)!+cfx"/(2n+1} w=YLeper ex®Amt+ox /2n=1)}} ‘Aconvenient rearrangement bystaggered summation results in w= Leff xamis cx"! (2m=)! oea) w=LH? {6.x/2m)!+e,x°*"'/ m+Di} since 7” Fo] = Wenow form theapproximants tothesolution u anifl= De and form theapproximate boundary conditions. Form= 1, 9(&)=b, and9(6,)=b, and form>1, 4 Paci(Ss)=By+[BrPal)dx Pani) =bs+[BrOa(a)dx ) (noting that timg,,, =limg,=u and that theapproximate boundary conditions become theexact boundary conditions inthelimit). Wehave9,=u,=ch)+xcl,Since9,(&,)=b, and@,(&)=b, ,wehave Uup(,) =b,andup(E,) =b, _——————— ad u,=eh)+xel—7 Fu, Py=Uy+Uy o.(E)=b +f°Bo(x)ox o(&)=b. +[7Ba.(x)x Since g,(3,)=b, andg,(&,)=b,, then 1)=[Bi(s)8x=f°B,vs(x)ox 1(&)=fFBondar=[*B,uolx)ex u,=02+xe?)-7Eu, =92-4 o,(&)=b, +[*B,0.(x)dx (Es)=bs+JPBeos(a)as Since ox(5))= bi+[7Bros(x)dx o&)=b, +[7B,(x)ex Q.= 0,70, then us(G)=ffBiws(x)ax uy=O) x0)" —7 Tues Pars =Px+Ue ani(&) =;+[/BPal#)Ex Gq-i(5)= Bs+fB.g2(x)dx Since 9,=Q,.-;=Us-) Inrecens Bovwonny ConorTIons 205 ua(G)=fFBrten(%)ex ¥&)= Badal Now using Uy=ch?+xc” usc excl" -yRu,, mel Thus u,=e =xei u,=oP+xe{)+(C7) x7/21+ eP?/31]} ugsof+xl+(=p)fol?x22/+2x74/(2n)(2m+DI} We can now write ef)+Ger =BI cy?+&,c =by? egge=BP) cf)+e) =by f+Ede)=v) of+ci) =De Forcomputational convenience, wedefinebi”=b,andbi”=b,.Form=0. Ye+iya(S(neler any+oe/(20+1)ex}OTS Snrleorslameosefan+0)}] ro-[afS ere toneeo9300/2941)jou —Ler" /C2nye+of"/2n-o4] a 206 Cuspren 8 Nowwecanwriteform20 oo)+,cf9)=BI? oP+Scf"=ve - which areasetofsimultaneous equations forthe matching coefficients c{)andc!™). Equivalently, fh) fee)_ fom)bratLd|7Lo} If&and&aredistinctastheymustbefortwo-point boundary conditions, omEb=ve)P=a8 bin? ple cm=tek sothat the solution isdetermined. ACCELERATION OF CONVERGENCE: ‘Wecanaccelerate convergence while minimizing further matching coeffi- cients bynow transposing from theboundary-value format, denoted byB.V.. toaninitial-value format, denoted byI.V. Theprocedure istofirstcompute a ‘currentbestestimate ofu('*”(thefirsttermofthedecomposition seriesinini- al-value formulation) thencompute u* form21 us)=(-7)°(E jus? he Yr)(Eees ulsg.fas 3(-7)*(0 un”? Inrecnus Borount Conortows 207 Forareasonable approximation tou‘Y” wehave galI=>(P(E )o,[eh] where g,.,[us"’]= "_,ul?=ull,Substituting, o,.[u=X(7(HPJou] where_ Ul=el el? Consequently, ueSenne) Ee+e] uh)=Y7)(IF Pelco] : +LC7)(Joule) The limits oftheseries fortheboundary-value solution andtheinitial-value solution arethesame, i.e., usye? 2he~ - B=Ball DCN +eaeENV) 5 which wecansymbolize bythealgorithmic form (sin7x) =Ga anil®4(GyulCol)*(Opa(C08 YF)+(Psuil6,D(} .{ol)-(®pa(c0s7) i ‘Then inthedouble limit 208 Cuarren 8 =i1fimPpseet] sin7x cy008.7x40, Wy which satisfies the original linear ordinary differential equation duldx? +7u=0 aswell astheoriginal linear integral two-point boundary conditions. MATCHING COEFFICIENTS FOR NONLINEAR INTEGRAL Conprtions: Forlinearequations, wewrote@,.,=J"Ut,a5an(m+1)-termapproxi- mation tou,i,forasufficiently high value ofm, Paai{u] =uorlimglu] =u (We note thatinpractice mdoes notneed tobelarge.) Foranonlinear func- tionofthesolution such ashu), weexpand intheA,:thus Paeilh(u)] =YA,{h(u))a which wecanwrite simply as"A, ifthecontext isclear. Thus h(u)=o,_.{h(u)] orh(u)= lim¢,,,[h(w)]. Theexactboundary conditions ae UG)=b, +f°Biai(u)ax ug,)=b,+[*Beh(u)dx The approximate boundary conditions are: ans{u(S)] =61+f-Bi@a[b(u(x))]dx eui[ul&)]=b: +fB,0,[h,(u(x)}dx regen Bovvoant Cowommons 209 which yield theexact boundary conditions inthelimit, Form= Iwehave g,{u(,)1=b, g,{u(S,))=b, Thecoefficients c™, c{*!arenowcalculable bydecomposition ofuandh(u). which means thatnonlinear integral boundary conditions canalso bedeait with analogously tolinear integral boundary conditions. SUGGESTIONS FOR RESEARCH: 1)Linear equation, linear integral boundary conditions: (f,andB,canbe constants orfunctions ofx) ux=5)=b, +iBy(x)u(x)dx ux=G)=b. +[*Boucods 2)Linear equation, nonlinear integral boundary conditions ulx=&)=b, +&By(x)hy(u(x))dx ula=&)=b, +[By(0)by(uC)dx 3)Nonlinear equation, linear integral boundary conditions ug) =b+fFReoucoer ulx=&)=b, +fBGdueodx 4)Nonlinear equation, nonlinear integral boundary conditions uta=§)=b,+FPBCn(ulx))ax u(x=&)=by+fBy(x)hy(u(x)ax 210 Cnarren & 5) Linear (two-dimensional) partial differential equation, linear contour- integral boundary conditions. Note thatC(x,y) =0implies x=&(y) Puldx? +Pu/dy? +7(x,y)u= g(x,y) Uafeyayne =.+f,Bmwude UODeanna Bs+f,Brayuae 6) Linear (three -dimensional) partial differential equation, linear surface- integral boundary conditions. S(x,y,z,) implies x= &(y.z) Fuldx? +Fuldy’ +HPuldz? +7(x,y,2)u= x,y,z) URY.2sayaa=By+§,8(x,y,z)uds URSDWsscapane =b:+$ Bosy.z)uds REFERENCE 1) G.Adomuan, Nonlinear Stochastic Operator Equations, Academic Press (1986) SUGGESTED READING 1)G.Adomian andR.Rach, Analytic Solution ofNonlinear Boundary-valve Problems inSeveralDimensions. J.Math,Anal.Applic.174,(118-137\1993). 2. G,Adomian, Partial Differential EquationswithInvegralBoundaryConditions.Comp. Math. Applic.. 9,(1983) CHAPTER 9 BOUNDARY CONDITIONS AT INFINITY Solutions ofproblems with boundary conditions involving alimit atinfinity canbedifficult. Tobuild some intuition, wewill begin with asimple example modelled by alinear differential equation. Consider the function use"=~,(-x)" /m,Weobviously haveu(0)=Iandu(s)=0. Bythe later, weclearly mean thatlimu(x)=0,Thisfunction satisfies thedifferential equation d?u/dx? —u=0.Letting Ldenote d*/dx", wehaveLu—u =0which isinourstandard format Lu+Ru=0with R=-|.With L”defined asa two-fold indefinite integration operator, wehave L'Lu =Lu sothat u=C,+C, x+L"u. (Since wearedealing withalinear ordinary differential equation, double decomposition isunnecessary.) Weidentify uy=C, +C, x u,=L"u, ~ u,=L"y, ug=Lug, =P ty =C,x™* /(2m)!+ Cx"(2m+)! Theresulting solution isu=\,u,,or u=C,coshx+C,sinhx Even from thefirst-order approximation 9,=uy,itisclear thatC,=1 since u(0)=1.Wealsohave thecondition u(se)=0 which might make usjump to theconclusion thatC,=0.However, wewould soon seethatwedonotthen geta verifiable solution. Since umust approach zero asxo», wehave lim{cosh x+C,sinhx}=0 sothat C,fimsinhx=~limcoshx or aun 2 Cnarren 9 Jim cosh x C,=-4==——=-limcothx=-1 Jimsink x >> andweseethatu=coshx~sinhx.Substitution oftheexponential forms cosh x=(e' +e")/2 sinh x=(e+e")/2 shows thatindeed u=e™*which westarted with, Equivalently, since coshx=Eo/tom sinhx=x7**)[omy we have= w=(1-x7/2!+x°/4!~ ...)-(x+x°/3!4x°/5!+..) sl-xex7/2t-x/3!+..=e7 Itisinstructive towrite C,==fim&5(x)/E,(x) ==lim7(x) where &.= cosh x bl<e £.= sinh x Ice 7(x) =coth x Ix]<7fortheseries representing coth x EXERCISE: Write theseries forcoth x.Using thePadé transform, show that thelimit, asx+©,is1.(See Appendix 1.)Forproblems inwhich thede- composition series isnotrecognized aswedidabove, wecomplete thedivi- sion, €.2., (x)=1-x7/2!+ x‘/4!-.-HOS TaeBIE RB then determine thefimit with thePadé approximant. EXERCISE: Consider theequation d?u/dx* ~p(x)u =0with u(0) =1and u(+ee) =0andverify thesolution. Bouvoany Conorrions arneiwery a EXAMPLE: Consider ageneric second-order linear homogeneous differential equation with boundary conditions specified atinfinity G?u/dx? +a(x)du/dx +B(x)u=0 LetR=odidx+B a(x)= Sax B(x)=&Bax" u(0)=1 Jimu(x)=0 wehaveu=)\, u,withu=C,+C,x. Then ug=(-L™R)® uy=(-L"R)*(1)+(-L"R)"C,x w=YCLIR)?-1+C, CLR)? x which we will write w=G(x) +C,8(x) Since lim u(x) =0we have7 =—Fim22)tim7¢x sin3)=m7) (x)= >rx (6=&.) ead G(x) =DGfx a . Sao 7(03)= Dtax= a yee 24 Curren 9 The series for(x)willusually haveafinite radius ofconvergence. Therefore, inorder toevaluate thelimit at+©10compute C;,wemust transform these- riesfor(x)oratruncated seriesapproximating ¥(x)toarepresentation suit- able atinfinity. The Padé approximant isuseful here. Write W(X)=Yo+7X472x?+...andcalculate T(x)whereI(x)isthePadéap- proximant of(x).(SeeAppendix I)ThenC,=~limT(x). MODIFIED DECOMPOSITION AND CONDITIONS AT INFINITY: Consider d*wdx’ -u=0 withgiven conditions u(0)=1andu(+©)=0 using themodified decomposition. Insome cases. itispossible that theresult- ingrecurrence relation could provide insight andilluminate thematching ofthe solution tothecondition atinfinity. Afinite decomposition approximant such as%=S._.Ye withafinite radius ofconvergence might betransformed intoafinitefractionwhichisaccurateforlargextoobtainthelimitatinfinity. Ofcourse. other techniques may beapplicable such asEuler transforms, ana- lytic continuation, etc. Ifwerecognize well-known functions asinour first ex- ample. thisadditional step will beeliminated. SOLUTION: L"Lu=L"u whére L"'()=C,-C,x=12() where 1,0=Jax Wehaveu=C,+C,x+Iiu where weletu=7,a,x”instead of SE. te.Then 2< agp Kus) —t2_ x= *x(m-1)(m) Consequently, ax"=C,+Cx+ 5238 yo +2Tama sothal we have recurrence relations forthe coefficients: Bouvoaky Conor: ativeinire us a=Cy a,=C, a,=—2at =~m(m-1) sothat a,=C,/21 a,=C,/3! a,=C,/4! a,=C,/5! Wecannow write a.,=C,/(2m)! anda,,,, =C,/(2m-~+1)! sothesolution can bewritten as w={Cox"*/(2m)!+C,x*"'/(2m +1)!}emt u=C,>)x*/(2m)+C,¥) x*"'/(2m+1) a End Atthispoint, werecognize thesummations; butletusassume thatwedonot see that these areseries representations ofthe hyperbolic trigonometric functions orknow their limit values atinfinity, andmust proceed inaway which isusable when theseries arenotrecognized. Evaluation atthezero boundary u(0) =1requires C.=1.Toevaluate atthesecond boundary gives us limu(x)=Oor Y)x*/(2m)!+C, limSx"'/(2m+1)!=0 ae a0 a0 hence Yx*/2m) C,=~fim}=*=*————_—. YPyem+y! days?c=-tim{1,2-* 42e_ey. rom [x 3 45” 945 4725 216 Cuarren 9 EXERCISE: Verify this series. Itcanalso bewritten yeeC,=-lim Lycon? Bewelx Sj Qm) where theB,aretheBernoulli numbers” .Welist, forconvenience ofthe reader, B,= 16 B,=6912730 B:= 1/30 B,=7/6 By= 1/42 By=3617/510 B.= 1/30 B,=43,867/698 Bs= 5/66 Byo= 174,611/330 The radius ofconvergence ofthisseries is7.Thus toevaluate { sonstim/t 8-228] xor[x 345° 945 J ‘weneed totransform the series, ofatfeast, transform the truncated series to representation which willbevalid asx++2 EXERCISE: Show, using Padé approximants, that thelimit of 1zee x 3 45 945 or LoS ye2 tet2+¥Cy"—B,x*Xast (2m)! is1asx. (Hint: Ignore the1/xwhich obviously does notcontribute.) Write thePadé approximant [2/2] andsince C,isequal tothenegative ofthis limit,showthatC,isequal0—1,Hence ”H.B. Dwight. Table ofIntegrals andOther Mathematical Data, 4thed..MacMillan and Cou. NY-(1961 Bouvosky Conomions atIweinry 27 us x2/2m!- Ye/2m+ Y={1+ 21+x'/4t+--+} -{xtxBl+ S14} sloxex2! xeBI+--= DY(-1)*x8/(m)! Thus thesolution iscomputed. Ofcourse, itistheexponential function and it iseasily verified bysubstitution. Weobserve that itmay benecessary totransform thesolution series toa representation forwhich wecaneasily determine thelimit asx °°.Auseful technigue isthat ofPadé approximants which allows ustoevaluate the matching coefficients C,and Ci, i., the“constants ofintegration,” (particularly thecondition atinfinity). Ofcourse, since wedorecognize the functions wecanusethefactthatthelimitasx>©ofcothx=1andthatthe limit ofe™*iszero. But ingeneral, thefunctions arenotwell-known norwill weknow apriori their limits atinfinity. EXAMPLE: Consider thecase d*u/dx*—p(x)u =0with u(0) =1and lim ua)=0.Letp(x)=7,pgx.WriteLu={S7 0,x*}uwithL= ldx?.Operating onbothsideswithL'=ff()dxdx, u=C,wcxet'{Zp, e}uea,x*ot Ext a= a= Ee} -~2 a ae L'=L' 7 =-y ——_ PixueLYx2Ptae DdTaxtmey oyPte = ea “2mma) piPbate =U Pens =C,+Cx+ a uxxees) Therefore 28 Coapren9 ay=Cy a,=C, andfor m>2 a © Peters a= 2°" mm) sothat A,=PpAo/1-2 8,=(pp&+p; &)/2-3 2,=(Dp8+P, 81+ Pz@p)/3-4 8g=(Ppas+P,Ay+PzA,+Psp)/4°S Consequently a,=Cy a=C, a,=p5Co/2! a,=pC, /3!+p,C,/3! a,=C,/4l+p,C,/3-4+ p;Co/3-4 a,=piC, /5!pyp,Cy/S! +P;PoCo/2-4-5+p2C,/4-5+ psCo/4-S Thus, va. {1+p,x?/2!pyx’/3!+(p; +2p,)x*/4! +} =C,{x+pyx?/3!+2p, x*/41(pi +6p,)x°/514 -} Sinceu(0)=1,C)=1.ToevaluateCywewrite 1=pox?/2!pix‘[414hd) C.=-limgy Teta raefatpyx'/3!+2p,x4/4!+ pax’[St+ Loox 3x? \C,=-Iim\+ +p,4-p2% +---) snlPoxPig ty) whichisthesameasourearlierresultifp=1,91=p:=0.Ofcourse,if p=D2, pax®wemustinclude theneglected termsofpintheratioforC, Bounoary Conorions arIneinire 219 EXAMPLE: Linear Third-order Equation: The function u=e*satisfies the equation d’u/dx’ +u=0with conditions u(0)=1,u(=)=0, andu’(~) =0, Letusinvestigate how todetermine this solution ifonly theequation and conditions areknown. L=d°/dx’ andL”isnow athree-fold integration. From Lu=-uand L"Lu=—L"u, u=C,+Cx+Cx72-L'Y u, where “ u,=C, +Cx+Cx/2 u,=-L"u, u,=-Ltu, =(-L" Puy u,=(-L")®u, =C,(-1)® x"/(3m)! +C,(-1)" x""3m +1)! +C,(-1)*x'*"7/3m+2)! Sinceu=S\, Uy,wewrite w=Co(x) +C,6(x) +C 64x) with 6-5corefom = &=>corefomens a &=>(1x?fama a Thus wehave w=Coba(x) +C,G(x) +Cz64(x) ul=CoSox) +C(x) +CrE(x) Since @=uymust satisfy u(0) =1,weknow Co=1sothat u=G+C,6+C.6 w+ C+ C& Since uandu’must approach zero inthelimit asx»e,wehave 20 Cuarren 9 Fim{E,+C,& +C,g,}=0 tim{6+C,6+C,&}=0 or &:)(¢) () tim) |<|==tim ole alas Ale Solving fortheCy.Czandwriting D=& &~6, (e)--umd ~&),(% (GJ DUS) Cy=~fim(E:E_~6:85)/D C,==fim(-E, ~8:65)/D Symbolizing thisasC,=fimp(x)andC,=~limo(x),wecanobtain Padé approximants ofpand6togetthelimits, EXERCISE: Show that C,=-1 andC,= Isothat u=e* EXAMPLE: Nonlinear 3rd-order equation—As anexample weconsider the Blasiusequationofboundary layer theory: Puldx?+(1/2)ud" u/dx?=0 orinourformat Lu+Nu=0with L=d'/dx’, Nu= (I/2)uu”, L"'defined asa iple integration andtheA,calculated for(1/2)uu”. Applying decomposition, usa+Bx+yx2-L'y A, and“ uya4Bx4yx7/2 The given conditions areu(0)= 0,u'(0)= 0,and u’>Ieu—++e.Thefirst wo conditions require a=f=0sothat Bounoak? Conomons artnewry 2 uy=7x/2 u,=“LA, =-(1/2)L"(ugug) =-(97/2)(x*/5!) uy=-LA, =-(1/2)L"(uug +uguy)= 11(7°/4)(x°8!) u,=-LA, =-(1/2)L"(uzuy +juluguy)=~375(7°/8)(x"'/11!) Thus, w=x7/2—(77/2)x°/5!) +11(7°/4)(x°/8!) —375(7*/8)(x"/11) which istheBlasius series! Now, w=7x(PRIN +A) => We note that u%(0)=7 and we have the remaining condition u(x) as xe forthe evaluation of7. Sincetheu’(x)serieslacksthefirsttermin™,c,x°,wetransiate, using x=z+§ with &taken as1/2,sothattheseries foreach new coefficient will converge veryrapidly. Wethenhaveanumerically equalseries~,b,2* with non-zero bg,b,,b;,... andcanfindthelimit oftheseries. (See Appendix 1)Since this involves 7and wehave u’(ce)=1, wecanevaluate 7directly without useofnumerical methods such asshooting techniques. EXERCISE: Carry outtheevaluation andverify thesolution u(x), Transform f(x)=7,©,x*wherec,=0,(usingz=x-with€<1andwithinthe radius ofconvergence) to)", b,2°withb,#0.IftheBlasius problem is given as an initial-value problem, we have u”+uu”=0 with u(0) =u’(0)=0 andu"(0)= 1.[1]The given conditions areu(0) =u’(0)=0 andu"(0)=I.Thefirsttwoconditions requirea:==0sothat uy=7x/2 uy=-LTA, =-(1/2)L"(uguy )=77°15! u,=-LA, =-(12)L"(uyuy +upuy’)=1x8! uy=“LA, =-(12)L"(uzug +ujuftugu)=-3757*x"/11! m2 Charter 9 Afour-term approximant ®=Din.yUsisgivenby w=7x°/2(7? /2)x*/S!+11(7° /4)x*8!-375( 74/8)x"LL Now, way PxBie ‘Wehave theremaining condition u”=1which gives usy=1. EXERCISE: Substitute theresulting solution, carrying terms through x°to seethatthedifferential equation andgiven conditions aresatisfied. EXAMPLE: Consider thenonlinear 3rd-order equation @u/dx? +mud?u/dx? -(du/dx)? +a=0 for0<u<ee andgiven conditions u(0)=0,u'(0)=f,andu’(ee) =7.We let L=d’/dx’,L"' isdefined asa3-fold integration, and wewrite CZ,Aa{uu”} forthefirstnonlinear termandS™,A,((u’)*) forthesec- ond. Bydecomposition, usu-mL'DY A,{uu}+L'S Afue’) uy=+qx+0x7/2-ax'B! u,=mL'A,{uu"}+L"'A,{u'u’} u,=mL"A,{uu"}+L'A,{u'u’} Wecannowformann-term approximant g,="|,u,,whichconverges to uasnapproaches. Because ofthecondition onu(0), wehave =0and becauseofthecondition u’(0)=8wehave n=f.Evaluating theA,for n=0 A,{uu”) =(Bx+ox7/2 -ax?/3!)(o -ax) Aj(u'u’) =(B+ox-ax!" Hence, BounoaayConomions arirtvere 223 u,=mb"{(Bx+ox/2-ax’i(o-ax)} +LB+ox-oxt2} EXERCISE: Determine thePadé limit and useittoevaluate theremaining unknown constant &..Verify thesolution toru, EXERCISE: Consider thesolution ofu”+u=0 using “modified decompo- sition”. Letu=307a,x™ andLu=-u with L=d'/dx’. Since uy=Cy+Cx+C,x"/2 (where C,,C,C, aredetermined from thespecified conditions), show a=Cy a=C, [on aS ag.) =—— 5 =n+ 1(m+2)\(m+3) form= 0,1,2,... « EXERCISE: Show that the solution is u=C,>(-D*xfomec. cote fomen: a ob +O)core? ome2 EXERCISE: Using thesame conditions aspreviously used forthe decomposition method, i.e.,u(0) =1,u(c=)=u’(es)=0,showu=e™*. SUGGESTED READING 1. R.E. Meyer, Introduction toMathematical Fluid Dynamics, Wiley-Interscience as7D. CHAPTER 10 INTEGRAL EQUATIONS Integral equations ofVolterra type arise quite naturally inphysical applications modelled byinitial-value problems. Consider thelinear Volterra equation ofthesecond kind, (Fredholm equations ofthesecond kind which areassociated with boundary value problems forafinite interval [a,b], are similar except thattheupper limit isb.) Ox)=f)+A]Ke.yoyey withaSx,ySb,andlet2=1.Using decomposition g=~,¢,(x), we identify @,=f(x) assuming £(x)# 0,then o(x)=0+)K(xy)>o,(y)dy andwrite=, @,asthesolution, orwithanm-term approximant ©=D2)gq.Insomecases,exactsolutions aredeterminable. Consider an example: K(xy)=ly- x} fie x Then a=Jly~sooty)dy= Jly-lydy=-x' /3! Thus thetwo-term approximant is, 979, +9,=x-(0/3!) or@=sinx asiseasily verified either bycalculating more terms orby substitution. Several illuminating further examples onconvergence appear in[1]. 1)Consider theequation 26 ran Eavarions 2s (a)=2+Jxeucoat Yvx(x)=uy +fxt,u,(tyee 2x u,=2%3 w=fixar=92g] WS 33hay 2x=3%OF vt 2x w=xfbar= 3 axl artte By us)=x O°Ge Thesequence=QyLitaDY[s)<l+s+ct seat4G 379° 277 8 Thepartialsumsare ,=1.000 8,=1.333 - 5,=1.449 $,=1.481 S,=1.493 andthelimS,=1.5. Then thesolution oftheproblem is: ~ => ~u@=F u@=> 22H Lex= oo 3a 3 2)Consider thenonlinear integral equation wa)=[|«=vorSkat 226 curren 10 Weget u(x) =0.75x+0.20 U(X)=Ay=ff(X=)uf(t)dt=0.23x-0.19 2,0)=A,=3f)OHu(t)COat =(0.00354712)x —0.004464994 ‘The approximant tothesolution with three terms is: u(x)=ug(x)+u(x)+uy(x)=0.98x+0.003 whichisveryneartotheexactsolutionwhichisx. 3)Consider anonlinear biological problem: Find arealfunction udefined onR by u(x)+0.25x{"Kx,2)g(ult)ét=1 with K(x) =I/x+t andg(u) =1/a.Thesolution bydecomposition is: u(x) =0.25x (log(x+1)~logx) athree-term approximant results in x os Ol 1.059947 0.2 1.089588 0.3 1.109975, 0.4 1.125276 0.5 1.137327 0.6 1.187124 0.7 1.155278 08 1.162186 0.9 1.168123 1.0 1.173287 Ofcourse, wecanimprove theapproximation with more terms until thede sired accuracy isachieved. Iwreaeas Equartons 227 4) Find areal function usatisfying theintegral equation: uGs)=Af}xtfucofat+x where 7.isareal parameter, A(0,1]. With afive-term approximation, decomposition gives thefollowing results: u(x) =x ua)=A,=Af!xtu(of?at=4% :2=AY,thug 7 af #x u,()=A,=24f)xtuo)woe 1 > SRyQo=A,=af) x[4u,(du,()+2u)@) a=ASx 1 U0)=Ay=Af!xf12(u,(0),(0+4,u,(O)]o= ax u(x)=14342, 5298Ix.4° 8° 22° 16 Assuming avalueofA=1/10,[1]showanerrorof9.7x10“~107. REFERENCE 1. Y,Cherruault, G.Saccomandi, and B.Some. New Results fortheConvergence of Adomian's Method Applied toIntegral Equations, Math. Comput. Modelling, 16,(85- 93) (1992). SUGGESTED READING 1.G.Adomian. Nonlinear Stochastic Operator Equations, Academic Press, New York (1986) 2.B.Some, Some Recent Numerical Methods forSolving Hammerstein's Integral Equations, Math Comput. Modelling, toappeas. 3. B.Some, ANew Computational Method forSolving Integral Equations, submitted for publication, CHAPTER 11 NONLINEAR OSCILLATIONS INPHYSICAL SYSTEMS Nonlinear oscillating systems aregenerally analyzed byapproximation meth- odswhich involve some sortoflinearization. These replace anactual nonlinear system with aso-called “equivalent” linear system andemploy averaging which isnotgenerally valid. While thelinearizations commonly used are adequate insome cases, they may begrossly inadequate inothers since essentially new phenomena canoccur innonlinear systems which cannot occur inlinear systems. Thus, correct solution ofanonlinearsystemismuchmore significant amatter than simply getting more accuracy when wesolve the nonlinear system rather than alinearized approximation, Ifwewant toknow how aphysical system behaves, itisessential toretain thenonlinearity for complete understanding ofbehavior despite theconvenience oflinearity and superposition. Physical problems arenonlinear: linearity isaspecial case just asadeterministic system isaspecial case ofastochastic system. Inalinear system, cause andeffect areproportional. Such alinear relation sometimes occurs butistheexception rather than therule. The general case isnonlinear and may bestochastic aswell. Insuch cases. itisnatural tomake limiting assumptions—which isnotalways justified. Using decomposition, these become unnecessary even forthestrongly nonlinear case and thecase of stochastic (large fluctuation) behavior, aswell asinthe cases where perturbation would beapplicable orinthelinear and/or deterministic limits. “Smaliness” assumptions, linearized models, orassumption ofsometimes physically unrealistic processes may result, ofcourse, inmathematical simplicity butagain may notbejustified inallcircumstances. Here weareconcerned with thestudy ofvibrations, orequivalently with oscillatory motion and theassociated forces. Vibrations can occur inany mechanical system having mass andelasticity. Consequently, they canoccur in structures and machines ofallkinds. Inproposed large space structures containing men and machines, such vibrations will result indifficult and crucial control problems and also lifetime orduration considerations, since vibrations can lead toeventual failure. Oscillations canberegular andperiodic, orthey canberandom asinan carthquake. Randomness leads tostochastic differential equations. In on Nowuwear OSciATIONS IvPursicaL SYSTEMS 229 deterministic systems—the special case where randomness vanishes—the equations modelling thephenomena orsystem provide instantaneous values foranytime, When random functions areinvoived, theinstantaneous values areunpredictable anditisnecessary toresorttoastatistical description. Such random functions oftime, orstochastic processes occur inproblems, for example, such aspressure gusts encountered byaircraft, jetengine noise, or ground motion inearthquakes sothatFmay beanonlinear stochastic operator inthemost general case." Insuch cases, wewrite FusLu+RutRy+Nu+Nu=g where thescript letters indicate stochasticity. Still more generally, Numay bea function ofu,u’.... aswell, butthiscauses nodifficulty. Inanycase Nuand ‘Nucanbewritten interms oftheA,.Although convergence ofthedecompo- sition series forywill bemost rapid when weinvert theentire linear determin- istic operator, computation oftheintegrals will, ofcourse, bemore difficult, also since wewill notthen have simple Green’s functions. Wewill letL denote thehighest-order linear differential operator. Inanoscillator wehave generally anexternal force ordriving term x(t), a restoring force f(u)dependent onthedisplacement u,andadamping force, since energy isalways dissipated infriction orresistance tomotion. Usually thisisdependent onvelocity andwewillwrite itasg(u’). Ifwehave afreeoscillating mass m_on aspring with nodamping, wecan write mu”+ku=0 ifthespring obeys Hookes’ Law, ie.,assuming dis- placement proportional toforce. Ofcourse nospring really behaves thisway. Often the force needed foragivencompression isnotthesameasforanex- tension ofthesame amount. Such asymmetry isrepresented byaquadratic force, orforce proportional tou*rather than u.Wemayhave asymmetric behavior butproportionality tou’.Then thesolution isnottheharmonic solu- tion which onegets forthemodel equation mu” +ku=0though itisstill a periodic solution. Thedamping force g(u’) maybeu”where cisconstant, or itmay bemorecomplicated suchasg(u’,u’*) soitdepends onvaswellasv. Byusual methods, analytic solutions then become impossible. “Whenthehighesdrvaiveappersinsalinetem,iplsoinsorbranches exist(1) 230 Cuarcen11 Suppose wewrite —f(u) fortherestoring force, —h(u’) forthedamping force, and represent thedriving force with g;theresulting equation will be + f(u) +h(u’)= g.Suppose therestoring force isrepresented byanodd function sothat f(u)=—f(-u). Wehave this inmost applications; itmeans simplythatifwereversethedisplacement thentherestoring forcereversesits direction. Apendulum, forexample, behaves thisway. Wemight take thefirst twoterms ofthepower series forf(u)andwrite f(u)=au+ Bu’. Then we have u”+au-+Bu’=g.Ifwehavedamping also,wehave u"teu'+ou+Bu'=g assuming thedamping force is~cy’. (This isDuffing’s equation [2}.) The simple case oftheharmonic oscillator mu”+ku=g,orthecase avoiding theassumption oflimited motion, hasbeen discussed completely in [3], and wewill consider more realistic cases here with damping and nonlinearity. Wemightnote,however, thatifinsteadofsinu=u,wegoastepfurtherandwritesinu=u-u’/3! wegettheDuffing equation with€asa small parameter, i.e.,aperturbation result, Itisclear then that wemay well have other nonlinearities thanu’.which weconsider inthefollowing sections The Duffing osciffator inarandom force field modelled by u’+au’+ Bu+yu’ =g(t)canbeanalyzed without limiting theforce g(t)to awhite noise andallowing af, tobestochastic processes aswell. The same applies to the Van der Pol oscillator modelled by u”+Guu’ -Su’~u= g(t),These equations areinourstandard form Fu=git) which can besolved bythedecomposition method [2-5]. Iftheequation is linear anddeterministic, wehave simply Lu=gorLu+Ru=g THE DUFFING AND VAN DER POL OSCILLATOR EQUATIONS AND REAL-LIFE PHYSICAL PHENOMENA: Suppose wemake measurements inthelaboratory andobserve afunction f(u) in a“Duffing” experiment and find an odd function f(u)=b,u=b,u? +b,u* +...orf(u)= ™, bu"! asshown inFigure 1 [NowuBiE4R OSCILLATIONS IvPaYsicaL SYSTEMS 231 fu) HFu Figure | or, on the other hand, we observe ameasurement ina“Van der Poi” experiment which yields aneven function f(u)=b,+bu*+b,u‘+... or f(u)=D2,b,u™asinFigure 2. flu) fe. Figure 2 IntheDuffing case ournonlinear oscillator equation is, u”+oru’+f(u)=g(t) m vu’+ou'+ Ybu =g(t) or uw”+@u’+byu+bd,u? +b,u?+--+=g(t) andifweretainonlythefirsttermofthesummation, uv+au'+Buryw=g ‘Thus equation (1)subsumes theDuffing equation. Similarly, wecanconsider anoscillator equation with nonlinear damping subsuming theVan derPoloscillator equation: 232 Cuarrer 11 u”+f(u)u’+Bu=g(t) w+{&butferss=a(t) u”+{by+bu+b,ut+--}u’+ Bu=g(t) Ifweretain only thesummation ton=1,wehave theVan derPolequation vu"+au'+Bu+ yuu’=g(t) with u(0)=cy and u’(0)=c,. Weseethat these equations, ascommonly used,aresimplyfirst-order perturbations ofthe real physical models. ‘Mathematics hasprogressed considerably using linearity andlinear operator theory. Nonlinear differential equations derived forphysical phenomena, ¢.g., inelectronic devices, have utilized perturbation theory orlinearization ofactual behavior. This issopervasive inthetraining and acceptance ofwhat is possible thatmodels ofphysical phenomena may beoversimplified under the assumption that considering thewue behavior will represent serious difficulties intheanalysis. Itishoped thatthedecomposition method may contribute tothe development ofmore sophisticated models and result inphysically realistic solutions tofrontier problems. DIFFERENTIAL EQUATIONS WITH EMPIRICAL NONLINEARITIES: Nonlinearities which arespecified only through experimental measurements then require curve-fitting techniques yielding series representations. Wewill consider ageneric anharmonic oscillator (subsuming thecases ofDuffing and VanderPoloscillators [2}) vu"+a(u,u’)+B(u)= g(t) withu(0)=e,, w(O)=c, gt)=D7,gt?with a(uu’) und Blu) assumed tobegiven asempirical graphs (i.e., asaplotied surface for aand a plotted contour for),Thecurve-fitting procedures result in: a(uu')= >¥a,uty” Blu)= ¥Bou* ‘Then theequation u”+a(u,u’)+B(u)=g(t) becomes w+dDYa,uru +>Bur=>et? Using thedecomposition and thepolynomials A,{f(u,v)}, discussed in Chapter 3,and thecustomary A,[f(u)] which wewill now callB,,wecan write (uu)=F,Ay(vyreeatyitynntt) Bw)=F,B(vynnt) Au)=>B[aw]=¥ sur=¥BEAlu] ie, B,[5@)]= ¥BoA,[e] “2zGandale)>A[u"] ie. Alau =X YDDoes Avfut} afer] where theA,,B,are aow specified. We can now use thedecomposition method towrite 234 Charen 11 Lu= g(t)- B(u)- a(u,u’) where L=4?/d1" andL"isthetwo-fold integration from 0totOperating with L",andsubstituting uu) —L1B(u)—L'a(uu’) uy=,+6,t+Lg(t) up. =-L'B, “LA, (m20) wecan writeu=~, u,andtheapproximant 4,[u]= 0,=D2)u,. Nowifweapproximate theinputfunction g(t)=J),g,t”bythemth order approximant oalgl= >B.t* then wecan compute thecorresponding mth order simulant tothesolution @,[u} oro,which satisfies theequation 9”+(04,02) +B(F.)= el8] 0,,(0)=c, 0,(0)=¢ ‘Thus thesimulanttothesolutiono,,istheresultwhenQq[g]isusedforthe input Tosummarize foragiven pre-set precision, weneed only approximate the input and thenonlinearities tocompute aconvergent sequence ofsolution simulants which approach thesolution more andmore closely astheseries for theapproximants arecarried farther. Obviously, thetechniques discussed can bevaluable insolid-state orvacuum tube electronics and device simulation The outcome should beuseful ingetting realistic models. The Van derPol equation, forexample, isassumed tohave theu’u’ nonlinearity andwehave used @(u,u’). Byusing the“best” empirical nonlinearity, weareinabetter position torefine themodel. Professor S.N. Venkatarangan (Indian Institute ofTechnology atMadras) and hisstudents have prepared several papers and dissertations nearing publication using thedecomposition concept. In[6]hefinds «closed form Nowuvear OsciuaTions wyPuvsicat SvsrEMs 2s solution fora(particular) Duffing equation byapplying aLaplace transform to thedecomposition series, then converting thetransformed series into a meromorphic function byforming itsPadé approximant andfinally doing the inversion. Thetechnique wasalso applied totheVan derPolequation andthe Rayleigh equation. Professors F.Jin-Quing and Y,Wei-Guang (China Institute ofAtomic Energy )have developed computer programs using thedecomposition method tostudyaccuracy ofthe solution oftheDuffing equation andforthefirst time tostudy chaotic behavior [7].The error isonly 0.0001% infourterms, which corresponds closely toourresults. REFERENCES 1. G.Adomian andR.Rach, Purely Nonlinear Equations, Comp. and Math. with Applic. 20, (1-3) (1990),2.G.Adomian,Decomposition SolutionforDuffingandVanderPolOscillators. Mathand Math, Sc...9, (731-32) (1986). 3. G.Adomian, R.Rach, R.Meyers. AnEfficient Methodology forthePhysical Sciences, Kybernetes, 20,(1991). 4. G.Adomian, Nonlinear Stochastic Operator Equations, Academic Press (1986).5.G,Adomian,AReviewoftheDecomposition Method,Comp.andMath.withApplic.21,(101-127) (1991), 6. _$.N. Venkatarangan andKRajalakshmi, AModification ofAdomian’s Solution for Nonlinear Oscillatory Systems, submitted forpublication. 7. F.Jin-Quing andY.Wei-Guang, Adomian's Decomposition Method forthe Solutions oftheGeneralized Duffing Equation andofIts Coupled Systems, Proc. ofthe1992 Int. Workshops onMathematics Mechanization, China Inst. ofAtomic Energy. SUGGESTED READING 1,V.S. Pugachev andLN.Sinitsyn, Stochastic Differential Systems, Jobn Wiley andSons (1987). 2.AM. Yaglom, Stationary Random Functions, R.A. Silverman, tran. anded., Prentice- Hall (1962). 3.VS.Pugachev, TheoryofRandom Functions, Addison-Wesley (1965).4.A.Blane-Lapierre andR.Fortet,TheoryofRandomFunctions, J.Gani,ransl.,GordonandBreach (1967). 5.J.Hale, Oscillations inNonlinear Systems, McGraw-Hill (1963). 6.A.Blaquitre, Nonlinear SystemAnalyses,Academic(1966). CHAPTER 12 SOLUTION OFTHE DUFFING EQUATION THE DUFFING EQUATION: Consider the Duffing equation with variable excitation and constant coefficients 0,B,7 u"+au'+ Buty’ =d() u()=c, —u'()=c, 6(t)willbewritten asaseries 6(t)="6,1? LetL=42/dt?. ThenL” will bethetwo-fold integration from 0tot. Lu=6(t)-au’-Bu~yu’ Operating with L*. L"Lu=L'6(t)-@L"'u’~ BL'u-7 Lv’ u=u(0)+1u’(0)+L6()- aL’ -BL'u-yL'w Replace ubyS",u, andthenonlinearity u’by"A, Wehave Ag=u A,=3uiu, Ay=Sulu, +3ujuy A,=u)+3uju, +6u,u,u, (A, through A.) arelisted inChapter 2forreference.) Aconvenient zlgo- rithm which also gives usthecorrect result forthisspecific case is mm Weidentify SouvtiowoFneDurrincEQusTiON 237 uu,=u(0)+tw’(0) +L"5(1) u,=-aL\(d/dt)u, -BL"u, -7L"A,(u,) u,=aL"(d/dt)u, —BL",-7L'A,(uy.u,) Ug.=-@L"(d/dt)u, -—BLuy-7L'Ag (Uosa) Thesolution istheconvergent seriesDu, andg,=D2yu,isthe approximant tothesolution. Although ourdefinition ofLasonly thehighest-ordered derivative rather than theentire linear operator avoids difficult Green’s functions, westillhave integrations ofthefunction 6(t)andtheA,.If5(t)isafunctionsuchas sin(1, thenonlinearity results inathird power ofsinqt, andweseethat a proliferation ofterms and computations can occur. (see page 254). We observe, however, that wedon’t calculate u_butarapidly converging approximant g,.Since thisisthecase, weneed notuse6(t) butan approximant ofitsseries, or,9,[5]= )\-",6,t° .Thecorresponding solution iscalled thesimulant gu). Itsatisfies, inthisproblem, theequation OG+AG,+BO,+7Tq=Gald].(0)=¢, o%0)=¢, with¢,(6]=0278,t°.Ofcourse,lim9,(6]=5(t) andlimo,,(u)=u. Consider anexample with @=B=7=1 and 4(t)=c‘sinx+e™'sin’ x. Solution bydecomposition willconverge tou=e“'sinx. Butwechoose to approximate e*bythe terms uptobut not including thecubic, so e“=1-t+(?/2). Forsinxwewrite x.Now g=x-—xt+(xt?/2) and L''g=xt*/2 since wearedropping cubic terms andbeyond. Hence, uy=sinx—tsinx +L'g=x—xt +(xt?/2) a,=0 sowe have eto,= [ltZz) 28 Cnarren 12 forour“solution” orsimulant forourapproximant of6which, ofcourse, is ‘sinxtothesameapproximation asusedforthesinusoidal andexponential functions. If0,isthemthorder simulant, 9,(u]= [7] Ifwecarry more terms forg{6], wecanidentify correspondingly more terms ofc,{u] or,inthelimit, u=e"'sinx. Assoon aswerecognize, or think werecognize, aknown series, wecan verify whether theequation and conditions are satisfied. Ifwedonotrecognize theseries, wecanverify thatcomputation ofmore terms, either fortheactual solution uwith theexcitation 6,orforg_ful with theexcitation (4), yields results which have converged sufficiently forthe decimal places ofinterest, remembering that weareinterested inphysical problems. Wecanplotresults forg,{uJ, or¢,{u}, for m=1,2,3,..N to showtheconvergence andestablish asolution approximant toasufficient accuracy Wenote thatforaspecified mthorder approximant oftheexcitation 6,we obtain themthorder simulant ¢,,ofthesolution. Ifweapproximate to,and including. cubic terms, weget vu=o;lul={1-e$-£xX)-esinsU 23 3 i.e.,tothesame approximation with u,=u,=--=0, wehave u=e"'sinx Ifone choosestousethefunction6ratherthang,(6),andtheresultisnota known series. onedoesn’t simply keep calculating with thestraightforward decomposition: astopping rule isrequired, i.e., there isnopoint infurther calculation, with theproliferation ofterms which result from nonlinearity, if theresults arepast thenecessary accuracy. Ifweneed three decimal places and thesolution hasstabilized atleast that far, itissufficient, Consider now the example: y"+2y'+ys8y=e™ yO)=12 y')=-1/2 Suppose wefirstwrite e"*=1-31. Wehave Soumowor reDureincEquarion 29 yo=(1/2)(I-t+0 -) yssa'sy, -L'yp-8L'yg sinceAg(y’)=y5 jt yoabSy,-L'y,LA)- dt AD=Y Ay=3ya¥1 AL=3Y0Y2 *3¥0¥i AS=3¥¥s +OyeTY Ag=3yi¥s+3¥0¥i+O¥0Ys+3ViY2 Ay=3yaYs +6Yo¥i¥e +OVoyays +3¥iVs+ByiV Ag=Y24OYY29s+3YYatyoY +6YoYaYa +OYo¥iYs +3¥a¥s at 1 : 2-2 Sa(1-t+-P)-La(l-te =? vsLioali-t P)-L'S(l-tee-e) =8bLaat 6100?+--+) SatSS 23°44°122°202°22 12p)fettee =h(1-tee-r)eE EE Ee w=7 ier ere ye eye ee 272 2°92 2 1 ia atieretan.|..il2} afeten Substitution intothedifferential equation shows N=0so(1/2)e™ isthe (exact)solution,Alternatively, wecancomputemoretermsandgrouptogether terms ofthesame power. Ifweapproximate e~with another term ofthe 240 Curren 12 series,weget 1 2_patpa tfi-tee Pee *x 4) ee te te yat-feh Eee eeots42047122440 fe Pe’ Hee 8. 4 2°22 sothatthetwo-term approximation @,isgiven by o=ti-tee-2)e...2 Ty sowehave another term ofe”'. Finally, wecanusenumerical results with increasing nfor@,toshow that theresults areconverging tothesolution, i.e., there isnofurther change within theaccuracy ofourgraph ortable. Ifwecalculate theone-term approximant ©,=y; forthesolution using three terms oftheseries fortheforcing function, wegetthefollowing results aee| [_o[.soo[so| fa [4s2fas | [4 [sss [3s | Convergence totheexact solution will bebest iftheactual forcing function, or atleast more terms ofitsseries areused, Inpractice, @,willconsist ofvery few terms. Another example isgiven by u’tu’+u+u?=cost~sint u(0)=1 w@=0 Soumon oFTe Durrinc Euston aay The solution bydecomposition with theabove excitation is_u=cost, obtained more easily ifweapproximate theseries represented bytheabove excitation, (Those already familiar with theasymptotic decomposition method willsee immediately thatthesolutionfort>+»isalsocost.Hence,u=costisthe solution forallt) There isnoproblem incarrying outsolutions where anyorallofthe parameters c,f,7, aswell astheexcitation 4,aretime-varying functions. ASYMPTOTIC DECOMPOSITION FOR THE DUFFING EQUATION: Consider thecaseofunity coefficients forconvenience: u’+ususu=6 Since weare interested inthesolution as_t —©,weintroduce thenotation Q[u]=limu(t)=limlim9, fortheasymptotic solution. Ofcourse theg,must becomputed bythe asymptotic decomposition method fortheabove identity tobevalid. EXAMPLE: Constant 6 w=d-u-u'-u" =u, w=SA, Aj=6=u, Hence u,=6% Ay=-u,—uy -uy or 3uju, =—u, 36%u, =-5% u,=-64/3 A,=-u,-uy-uy 3uzu, +3u,uy =—u, 242 Cuarren 12 A,=-u,-uj-uy=0 3ulu, +6u,u,u, +u)=0 364u,+664(-6%/3)(0)— S'/27=0 36%u,=6°"'/27 u,=5481 9-65 9,=64-54 =9, 0,=54-6%3464/3! =6%—1/36% +1/354 et.ajyj=i-teb-._ “3781 IfB>>1. Qfuy=3% If5<1, theseries diverges. ‘Thus wegetresults for521 inthisapproach. Asymptotic decomposition is applicable tolinear aswell asnonlinear equations: hence, u=6-u'-u'-u" u,=68 u,=-A,-u,—uy u,=-u2 =-8" u,=~, “usu u,=~3uzu, =-367(-6”) u,=35° u,=-A,-us—uy uy=-3uzu,-3u,u? u,=-1267 Now Q.=5-8 +36 -126" andwenotice thattheseries converges for0<6<1 SouwtionoFTHeDuFFiweEquarion 13 Itappears therefore thatthemagnitude oftheexcitation must beconsidered foraconvergent result. Thus ifu”+u’+u+u’=6 for0<d<1 wesolve forthe u, ie. wewrite u=6-w’—u’—u"; while if5>1, wewrite w=§-u-u’—u”. Thus wehave achoice inasymptotic decomposition which wecanusetoadvantage toobtain aconvergent series. (We also notice thatif6(t)=sintwhichisbetween 0and1,Qu]behaves likeaFouriersine series.) STAGGERED SUMMATION TECHNIQUE: Let’s first consider the harmonic oscillator with variable excitation to illustrate procedure: u"+au=B(t) oconstant (0)=cg u(0)=¢, Assume B(t)= ",Bt Lu+au= Ait) Lu= B(t)- au u=u(0)+u(O)t+L" B(t)-L" au Uy=cy+et+ >B,&fasen(a=2) ‘Wenow will write upasaseries; thus, Fo where a)=cy a=a, aft,=B,/(n+1)(n +2) usu,-Ltau Withthedecomposition u=xu, 244 Cuarren 12 u=-L' eu, u,=-Lteu, u=-L au, u,=-L'@u,,=(-L"¢)* uy form> 1anduy=7,al0°,Hence eee ete ue=CN a"Fe =o TH(n+v) Thesolution oftheequation webeganwithisu=x,u,.Ifwelet 1)=CDPaae a? = T(n+v) wecanwrite w=", °*SO", al"t®.Wenowrearrange results by staggered summation asshown inthefollowing tabulation, Pawan Taree are To]Ca aa Eea waalealit Pa(aleagee[=aeat)eal} e(area)e|(aeat=aJe STEES | w=Su=S ae B78 Altcmatively, thesolution components canbecomputed asfollows: SoumiowoFTHeDuFriaEquarion 25 yes we . w=Lawuse Fe where -aal”) Ws. (a+i(n+2) we awet Se where qo) 2a *(aea+4) wt de ~aa? a?)=2s (n+5)(n+6) Continuing, wecannow write ase sae Fx} aae) (=oraf (a+2m=1)(n+2m) Since u=>, uy, a=FoF Fae ree which isthepreviously derived result butmay bemore convenient for programming. Bystaggered summation, wedae et Form=0 246 Cuarren 12 a=a? ay=a) For m> 1, ies Dyalas rod Breer =D,Mheetees ‘Wenotethatwecanwritethea!"intermsofthea!” -aa” a)=0a(a+i)(n+2) ave aay) aal *@r3+4) Tay) -aa®)—— ata) aeeal nea +508) Tine) aia)=CDEaEa?= T(n+v) asexpected. SUMMARY OF FORMULAS: wed ey ae mo where for m=0 al)=cy al=c, a?)e Be ne l(n=2) and form2 1 SoumonoFTHeDurricEauarion 247 : a= -aae)>*(+2m=1)(n+2m) Since aisconstant-valued “ayeg®al? aor Uta Ta +v) HARMONIC OSCILLATOR WITH VARIABLE EXCITATION AND SYSTEM COEFFICIENT: u”+a(t)u= A(t) a(t)=¥, ae B= >Be a u(0)=cy u'(0)=¢, L=d?/dt? We have Lu+a(tu= p(t) Lu=f(t)(tu - L"Lu=L" (t)-L" a(t)u(t) u=u(0)+u/(0)t+L" B(t)—L™ a(t)u(t) uy=u(0)+u/(O)t +L" B(t)=cy +e,(t) +L"Blt) Uy=,+et+L" >)Bt Fd Uy=cy+e,t+L"byastfeastKa2) which wecan writeasuy=Y,a”e*where 28 Cuarren 12 al)=cy a=c, oo2B.Boer(n+1)(n+2) The following components arederived from u=u-L' aS, a Ex} Thus u,=-L" a(t)uy u,=-L" a(t), u,=-L" a(t)u, u,=-L' a(t)uy,=(-L" a(t)" uy Since a=) a,c (See lFameleselt | (thu=|Dae|Yah =Yra, al}(ers Aes) US 2 hence = lS =F Saal ‘=ape ae) where a eat ens 2) u;=-L" a(t)y, Soumawor rt Durr: Egusno! ne @(t)u,={Saclle Sere}=eby“Se.«) Substituting, wehave —pee eft ole refs «|eeeEe(So.01} &weyey Be) waeFave where = (n+3)(n+4) Farle Face! (tu,“zeMe=ii .att“(Se«} Upon substitution wees “fS.oh a5 8 Sa aeywar {Eewo} yale we where. Sa... a? aa *“(n+5)(n+6) wants Babe where:~ 250 Cunrrex 12 -Sa,, a ae) = at * (a+2m—1)(n+2m) Finally, since udu wehavethedecomposition solution ueSo Fawe This result canalsoberearranged bythestaggered summation procedure: thus ueSage E] where for m= we have a,=al” asa’ and form2|wehave a= Yan, and . sees=DyAnat Fert SUMMARY: ued oS alee BS Form=0.wehave avec, a=, ao,-_4 2ae ljin2) and form>|.wehave SouwmonoFrieDuFFINGEquaTion 251 {2**(a+2m—1(n+ 2m) HARMONIC OSCILLATOR WITH VARIABLE EXCITATION AND A DAMPING TERM: u’+au’~Bu=7(t) Leta.bbenumerical constants and7(0)=~,7,t°.Specify u(0)=¢, w)=«, L=d°/dt? toget u=u(0)+u'(0)t +L"7(t)-L" au’-L*Bu = yet Uy=c,+c,t+ >,—2—__ °°2(n+i)(n+2) a where ay”=cy a=, ),=-——%a eae i)(n+2) Continuing, u=-Ltauy-L" Buy u,=-L" auf-L" Bu, usb! aul, -L" Buy, Sinceus=Ya, t*(n+1)= Y-,bf?e°where be=(n+ 1a, 252 Cnarren 12 wee swe Fd a0)=2e(n+ Na~Bal?_ab? ~Ba” ° (a+1(n+2) (a+(a+2) Thenext component uwisgiven by u,=-L'aui-L" Bu, and > ror uD (ne2)aM etary we where bf”=(n+2)al”. Thus, we Save weaty le = abe (epee weeSCoble LeFCBee=(n=2)(n=3) 35(n+3)(n+4) We can combine these terms towrite wet Sale if- arabe °QO) and af,220 =Bal! er (n+3)(n+4) Now going ontous, web au;-L Bu, Since SouionoFrieDuFFiNGEauaTion 253 wed erore’dFler wed (nese erae FEwe Et Et withby)=(n+3)a, = abt = gayewet See FBae “3 (n+3)(n+4) Sp(n+4)(n=5) wae he where al)=—a bj/(3)(4) and a),=eb Ba?(n+ 4)(n+5) noting thatb\~” =(n+m)a(®"”. Finally, wewrite we Fae Fd where al)=—-a bi?) /m(m +1)and a)-rab Bae? “(n+m+l)(n+m+2) Thus, form=0, Fd and form>0, ya de Fd wae Sage Fd Fx} 254 Cuarrex 12 ug= al (m>0) w=Suse uy uaF esy erFale a a MS which isthesolution bydecomposition, EXERCISE: For adamped linear oscillator described byd°u/dt? +2du/ét+u=0 with u(0)=aand u'(0)=0, show that thesolution for decomposition is 2 op ye ag tvealaSarstytef, ou "3 4a "Ss! 6 EXERCISE: Fortheundamped nonlinear (Duffing) oscillator described by Lu+u+u'=0with u(0)=a and u'(0)= 0,show thatthetwo-term solution is, rat’ pea athr ar EXERCISE: Consider u”+au’+Bu=yu’=g(t)withw(0)=a.u’(0)=0. Assume a=0.B=7=1.Forg(t)assume sintandapproximate with thefirst, two terms ofthe sine series. Show u,=a-sintea-t+r/3! u,=~a?+3a't+3at?—(1=a°/2)t* PROLIFERATION OF TERMS: Innonlinear equations, where theinitial component ofthedecomposition solution consists ofseveral terms, thenonlinearity may result ir.aproliferation ofterms andconsequent increased computation unless proper steps aretaken. Also itissometimes convenient towrite theinhomogeneous term asaninfinite series tosimplify integrations. Itmay well bethecase that theexcitation is known asapower Series representation. Weconsider such acase here since it Soumow orrue Durric Eauarion 255 isa“worst case scenario” from thepointofviewofproliferation ofterms. If ‘oneseeks thecomplete solution (steady-state plus transient), thenumber of terms ineach ofthecomponents u,,form>1canincrease rapidly because of thenonlinearity. Anumberofpracticalaltematives arepossibletosolvesuch apparent problems, and wewill show that rapid convergence tothesolution will beobserved. Summations areutilized toorganize thederived results. Wenow consider theDuffing equation with variable excitation andconstant coefficients . u+au’+But7u =g(t) We assume given conditions u(0)=cy andu’(0)=c, and assume that g(t)=xg,t”sinceitismoregeneral, willfrequently simplify integrations (e.g., ifg(t)=cos nat) and isaworst case from thepoint ofview of computational difficulty arising from theaction ofthenonlinearity ontheinitial term(u,=u(0)+tu’(0)+L""g(t)). LetL=a?/dt? anddefine Lasthetwo- folddefinite integration from 0tot.Wehave Lu=g(t)- au’-Bu-yw Operating with L"' u=u(0)+u‘(O)t+ L'g(t)-L'au’-Lpu-L'yu* Weidentify theinitial term Up=u(0)+u'(O)t+L*g(t)= u(0)+u'(O)t+L"Y g,t® up=uO)+u(Onr+ SsEE_ &(a+1(n+2) which we will write as cd where thecoefficients areknown: 256 Cuarrex 12 a=cy A=e, 0,= _—&2(n+1)(n+2) We can now write u=u,-L'@u’-L"Bu-L"yu’ and assume u=7,u,andw=", A,where A,orequivalently, A,{u’) is calculated forthefunction u’.(We canalsowrite both uandu’as sums ofthe appropriate A,;thenubecomes simply u=J”,u,andu’becomes (4/at)",, u,.TheA,polynomials foru?are Ay=us A,=3u5u, A,=32u, +3u7u, Ae=2YacsDYaastl Algorithms have been previously given togenerate the A,forgeneral nonlinearities: however, theabove form isalso convenient forpolynomials. We now have veu,-Lied uf-LiS w-Ly ¥a(u'} Ed Ft Ford and canwrite thecomponents uy=5a? u,=-L"au;-L"Bu,-L'yA,{u'} u,=-L'euy-L"Bu,-L'yA,{u'} SoumonoFTHeDuFrivaEquaTion 257 u;=-L"'au; -L"Bu,-L'yA,{u’} Since uyisgiven asapower series, wecanusethefollowing result: Ifw=, at,thenf(u)=7,Agewhere theAxa... a)are simply the A,polynomials expressed interms ofthecoefficients a, instead ofthecomponents uyu), Continuing, wecompute wu;andu,. uy>at uy=ai+atte =Sins =Ybor by=(n+ale, Ag(to)=Ao{u’}=u3 =>AgeEd using theabove result. Thus AQ=a} AY)=2,05+a3;a5 +a5, and ade{52,529,“|Fave Caer) Fd where We{E2,549a We can now write 258 Cuarrex 12 u,=-L'aus-L"Buy-L"y A, =L'a SD(n+tale’ -L'py ah’-L'yy Ar =-aD(n+Nate?jon+1)(n+2) BYar?|e+1)(n+2) 7DAore[a+1)(n+2) ford =D? {a(n+Nal?)+Bal+VAP} (n+In2) B Weseethatu;isknownintermsofthea”andthatwecanwritetheequation foru, inthe convenient form: wat da or5ale? where at=DEON =Bal= . (n+1)(n+2) uy=DS(ne2)ae t=ey)we? where bt=(n+2)a\"” Ay=A,(Wo.tt,)=34,u5 where =o weed ave Te = = 9 Aya3e| SalSaSal”fer Lnso ad med where SoumowoF me DuFriws Eauariow 350 a Fat) ya olaS Fo.| efZeeDaea?i Wenowhave wae Sale uae) pie Ayaby le and ~ osaoe! -bigu-Llya, <P&abeayo Liaus Liu -Liva=eSa = (pr (cleay (-B)ate ay (nevews(n+3)(n+4) a(n+3)(n+4) Thus. upae[reese_abiteatte23. 34 45 pa” Bal re+r{Ban_Bait_Bar’_ 34 45 5-6 Ac A oS 34° 45° «56° * 230° 34 4-5 ab!) 0 0 uae. carbs)|as]Bae_yehe23 34 3-4 pale veld? Oe yeeapfcbaterete), afaateredea5 45 56 56 wee Sate Et where 260 Cuapren 12 a)=obi =Ba?-ve) . (n+3)(n+4) Going ontous u,=-Llau; -L"Bu, -L"yA, Sinceu,=07,af1°",wehaveuy=0",(n+3)aleeor use? ype Eo where be)=(n+3)al” For Aswehave A,=3u,u} +3u,u where wae 5arr wat 5ale Hence ~[a . ) Ala30S85 alDala”| a3Defy al,Yall, al) which we write as A,=30 DYoth+3YwtFood Ed and finally Agate where. eff!=305=3a?(al”) and SoumowoFHeDurrmcEquation 261 of,=3{0,,,+ v,} Thus, wees abe wet yare aa’ dSae uy=-Lau! -L"gu,-L'7A, ae_aye? petyyet weed Gana we&aei(n+4)(n+5) 5net" oy>(aranz5) ‘ormoresimply, asbefore, wy=e ave Fd where (9)abs on s4 a),=Deb =Bal~7c? at (n+4)(n+5) Continuing touy,us... ,Wecanwrite themthterm ue ae 3 gncave >'m(m +1) ata)=ebey?—Bal? ~ye? ot (n+m+1)(n+m+2) Summarizing, form=0. 262 Cuarrex 12 y=8S et wee 5ae a wetdyae uae awe a Thedecomposition solution isu=~, u,oru=u,+>, uyor uy aes Seryaw ‘Wecanalso write thesolution bystaggered summation intheform usDar where ”asa? a,=a” and for m>2 anal +>al, ASUMMARY OF FORMULAS USED: Fae aEery awe Sowriaw oF rue DrFIG Equarion 283 (m=0) alc, al=c, co) Ba feet (nd) arb—Ba7 (m=1) al?=AO=Bal=rel(n+1)(n+2) able"(m22) a2)aem(m +1) ain)aU Bayer ot (a+ m+1)(n+m+2) where form=1. bi=(n=1)a\®, andform>1,b'*"?=(n-mal". We also have u=du, Eo wea, Aafu"}= DitaeDeYaote mS ALGORITHMS FOR THE DUFFING EQUATION: Weconsider now explicit solution oftheDuffing equation vu’rau’+Busyu"=5) giveninitialconditions u(0)=¢andu’(0)= ¢,.Weassume 6(1)= J,5,1°. Ourobjective isanefficient procedure forcalculation ofthedecomposition series. We have u=u,-L'au’-L“Bu-L"yu° where uu,=u(0)+tu’(0) +L“5(t) restorefflZara 264 Charren 12 Wenote thatiftheinput isasinusoidal function, integrations willsoon become difficultbecauseoftheeffectofthenonlinearity. Theresultfor6(1)=sintis found bysetting 5,,.,=(-1)°/(2n-1)! and6,,=0. Forcost,let5,,=(-1)*/(2n)! and6,,,,=0. For(t)=c,cost+c,sint, wecanlet5,=(-1)*cy/(2n)! and 6,,,,=(-1)°¢,/(2n+1)!. Finally, in many cases, afinite series willbesufficient fortherequired accuracy andwe canwrite£()=DNgt,ie,6,=f,for0Sn<Nand6,=0for n>N. Conveniently, 6(t), assumed tobeuniformly convergent, canbe programmed asanN-component vector with theseries truncated tothe precision required. Terms afer u,aregiven by uy=Laul, -L*Bu)- LyAn, wheretheAy..=Ag{w} =D S0juecleat: Ox=DegYsform>} istheapproximant tothesolution. @,.,=@_+U,. and limo, =u, Now write theinitial term u,intheconvenient form: a=Sa =9, Fe] where ali=cy, ase, a@,=d,/ns)in+2) Theapproximant ¢,willbewritten a=Dlr Fed where |=2{°. The u,component cannow becalculated andtherefore the@ approximant, since @,=@,+u,.Since uy=Dal’ wehave- uy=Dims bale Ed Sousmow oFTHe Dutra Euuarion 205 Computing A,using thegiven algorithm, we Bee Deaadenety Ay=u which weknow from ourgeneral algorithms forf(u) aswell. Using ourresult foru,,wecanfindA,astheCauchy product: or which we will write as y=SAME where“ Since u,=-L"'aus ~L"'Buy -Ly Ag,wecannow write =n oeame =Ame e-aP FS _gepy te FLAsaad POLerner 1Larned Insuccinctform,u,=~, al!®where a)=a(n+Da,=Ba-7Ae 7 (a+Den+2) Wenow have@,=9,+u,where Q,=LT,beandu,=eyTales hence: =Yee & 2668 Cuarren12 where b= bf? b=b b®,<b”), +20? Proceeding totheu,component andtherefore the@,approximant, u,=-Lau;-L'Bu,-L'y A, sothatfromu,=1?))" a*wewrite uy=e(n+ 2)al? and . ete AL=>Yuecue. =3upu, ortheCauchyproduct \e (Sool lak AaNYAM LSale’ eyalee AscDaie where “ Ag=3)) Yala? al? Consequenuy seat ate ae Ate upe-eP FE gy yep_At°dasa! Lana” Lane or waeSart wherea a?=-aa/3 ge=ainDal, =Bal?=7A? “ (n+3)(n+4) Now wehave theapproximant g,=, +u; with oneSoe wat Saee where. “ yaa wane Proceeding totheu,component andconsequently the@,approximant, we have u,=-L"au, -L“Bu, -L"7A, using warSaee w=Ya +3)aer A=3Ear}.{Eaor|. pear eSaor}-feSarr} {Ere} 268 Curren 12 which we will write as A=P Ae where - AD=3y Dal,aa? AM=3D Dalian aay? ce} 233Fa,a,2 Thus Balt pega u=-or'St gsy atmo(n+4) 8et(n+4)(n+5) AaKS -yty AtLae or wat Sar" B where af=aa?/4 at,=Ten4a,=Ba=7A“ (n+4)(n=5) Thenextapproximant 9,=9,+u,using =Yeen® warSale sothat . a=Dover Ea with by=b2), bi=bi",be=bY,by=dy,andbyt,=bP,+2(? Continuing totheu,component andthe g,approximant, SoumowoFTHeDurtwaEavarion 209 u,=-L'aus-L"pu,-L"yA, for which we need erp ag & wat dmedal ae AsDD veateatly A,=u)+3uju, +6u,u,u) Ay-{eSee} (eEsr}[egoe} alwe{5we}[oFvel +fwe}{eSwe}LPSwe A=Oded Fal, B48 HSe-3yFa2,a ae aS eyFa,a,a? which wewrite finally as. Ayer age where“ 270 Cnaoren 12 ap=ay,bya,al A?35.Daa2)a” +65 Fal. a,a? am ADS=3D avale? +dSYalat’a” Thus theu,component isgivenby mar alt u=-ary2 ges atara OSwar ce Noe ayyArt"% Grams which we will write as. wetae where- af=aa ° 5 ait=2a+Sal’, ~Bal’—yA’ ot (n+5)(n +6) andwehave thenext approximant @,= @,—u, with m=>dor weed alr SoumowoFTHEDuPriwcEguarton wt We write now => vor with “ Oe<b BP=a bab! BP=o bP=dy bly=byes+a,” Computing theu,component and @approximation. u,=-L' au,-LBu,-L'yA, for which we need wavy awe werd m+sair AL=DD teat te fed A,=3uju, +3u7u, +3ujuy +6u,u,U, A=fwe}{5or}eSa}B 4 = fedwel{o|foel Ed a = Fon] a = lSwe}{e§we}Fae = cd = Aad 0-35 Ya, 0,ery o3Ya, aa? SB Om a OM +Sv3EYaa0eSv6,Vaal’, a0?Some mB es 272 Cnnrren 12 which we now write as ‘ Ane aoe where- aa A=35) a®,a,a43) Ya 2,20 +65, Ya al,a” am ies 3 is u,cannow bewritten = a gt = wpeealt . al? . Aly u=-aty tgp et__yyyAct_ p»(n+6)idx(n+6)(0+7) D>(n+6)(n=7) and finally aay, ar® where“ “9_oatag=2 a,=Dal+Bays,—Bas! 7A a (n+6)(n+7) Thus.thesix-term approximant g,=,+u, whereSo"bv"and ug=U"alt? andweconveniently write => vor & with Soumow oFre DuFFING EquArION 73 be?<b b=” bi=D bi=p? oy=o? by 0? bY, =bd,=a Wenow have asix-term approximant tothesolution and, ifnecessary, can continue inthesame manner, (The algorithm caneasily beprogrammed for machine computation.) SUMMARY: Decomposition components aregiven by: w= ae? ugar yah mel Sogee = Thecoefficient-generating algorithms are: ay?=co a=, a,=6 (a+ D(n+2) 0=mln+Da,=Bal”=7A ° (a+1(a+2) o a)20a a@)= rr) a,=2a(n+ Bas,=Bay7A?eet (n+3)(n+4) eatag)=Sa m22 (m+D 274 Cuarrer 12 Ace ate = As=P 3are = ager Saoe Ssavew a 3 Pn=>,B® lim9,=u Pa cd REMARKS: Amethod ofapproximation does notneed infinite precision. We need toknow thatwehave uniform convergence ofinput andcoefficient series andgenerally 6(1)canbetruncated after afewterms forsufficient precision, Computation will investigate theeffect onincreased accuracy ofadding terms tothe 5(t) series todevelop stopping rules. Anexample isinstructive, Consider u”+u’+utu=g(t) u(0)=0 u'(0)=1 g(t) =1-(1/2)¢ +(5/6) The solution issint.Ifwecalculateu(t)bydecomposition. wefindg:=u(t) =1-(1/3)U orsinttothesame approximation astheseries for g.Ifwe specify ¢further, wecangetmore terms ofu.Practically, this means that if weareinterested inaccuracy tondecimal places, butfurther calculation results innofurther change inthenplaces, thesolution isconsidered known and will beverified tosatisfy theequation and thegiven conditions. Because ofthe generally rapid convergence, which wehave seen inallwork onthesubject, wewould expect that most ofthesolution isinthe early terms ofthe decomposition series. Ifwehave the @,approximant tothree decimal places and further change exists only inthefourth offifth place, ourresult may besufficienUy accurate sothatconvergence toarequired precision canlead toastopping rule. Aswe gain insight from ourcomputation, wecandevelop stopping rules, investigate Sournon oFTue Durrive EyusTion 275 convergence rate anderror, acceleration techniques (Padé, Euler, Shanks), checks onthesolution bymodified decomposition, asymptotic decomposition, and verification that thesolution satisfies thegiven equation and conditions, determination oftheregion ofconvergence ininitial-value problems and possible useofanalytic continuation, ourone-step procedure, andother special techniques. Because ofourchoice ofLandRwithRalower-order operator than L,wealways have convergence, aswesaw inevaluation ofmatrix inverses with [L"R| <1,andsince ourintegrations aretrivial. i...wehave Green's functions ofunity. Adomian and Rach (1]have previously discussed aninitial-value problem with asimple diffusion equation with u(x,0) =f(x) where theoperator inthesolution series u(x,t)= ),(LEL£00) annihilates f(x) atafinite m;theresult then does notsatisfy theequation and conditions. Noclosed form solutions exist fortheDuffing equation andsolutions have always been made using perturbation, discretized methods, etc. The results have shown multiple oscillations—not only attheexcitation frequency @but alsoatsubharmonics /n,orsuperharmonic nwwithn=1,2,.. Iftheexcitation contains several terms wegetcombination frequencies also. Thenonlinearity causes complex interactions among theinput terms: however, theseries captures alloftheactual result. One could plot theresult and doa Fourier analysis toseecomponents intheoutput. With non-conservative systems where theu’term isnon-zero, one would notingeneral expect sinusoidal outputs unless g(t)compensated appropriately forthedamping. We must becertain thatwehave acorrectly modelled problem whose solution is, consistent with thegiven conditions, andthen solve ittoseeiftheharmonics doexist infact. Finally, solutions areunstable against small changes which canresultfromround-off errororlinearization betweengridpointsintheusual computer methods. EXAMPLES: 1)Toshow that thedecomposition method yields correct solutions tothe degree ofapproximation thatweusefortheexcitation, wewill begin with u=sint=t-1/3! and substitute into the Duffing equation u”+u’+u+u’ =g(t). Dropping terms greater than 1’,wehave 276 Cuapren 12 -1-ie +Sv 8 2.6 L'g5(+higher terms>1*) u,=u(0)+tu'(0)+L"g e etekWyatt T web) y-be‘ 2 6 wen” toorderof award g ppp MT TR 0a5orsinttoorderofg ‘We now have thesolution tothesame degree ofapproximation asg.Thus, given theproblem u”+u’+u+u’ =g(t)where g(t)=1-1°/2= 51/6 with u(0)=0 and u’(0)=1, wegetu=sint tothe approximation g.=t-1°/3!. Asthe forcing function isapproximated more closely, our series foru approaches theseries forsintmore closely. Substitution oftheapproximant @,imtotheequation must satisfy theequation tothatdegree ofapproximation andsatisfy thegiven conditions aswell 2) Let - Sp Sep! =re yO yo geetesint=PT*Gash! u’=-0/2 and u”=-t, substituting inthe Duffing equation approximate uwiththefirsttwoterms ofeach series. Thus u=1-1°/3!, Then uvtu’+usu =g(t) Wegetg=2-1, since other terms aregreater than t'when operated on byL":Thus L“g=t?-1'/3!. Since u(0)=1 andu’(0)=0 SouvriowoFmeDurrieEguarion 277 uy=1+e z Togetu,=-L'u, -L“u, -L“u, eee =-L12-L"()-L"()=-+-+-> uy)-L'M-L'=-F- 5-5 °peop.3 uy=-2t- uy=0 (ie,>0) 2Pri, sie-L-bap%a3 e 9=1-> u,iszero tosame approximation sowehave =u, 2%-a-9+(1-£) ‘Thus,giventheproblemu"tusutu’ =2-1 u(0)=1 u'(0)=0 wegetg,=1-t/2 which iscorrect even ifwedonotseethatthisis (1-1)+(t-1/S!) andrecognize orguesse'+sint. A CONVENIENT FORM OF THE SOLUTION OF THE DUFFING EQUATION INASCENDING POWERS OF T: Considering theDuffing equation u”+au’+Bu+yw’ =s(t)where 0,B,y areconstants, u(0)=c,,u’(0) =c,and g=,g,t®,wecalculate thesolu- tion through thet*term which should generally besufficient. Letting L=d?/dt?, R=a(d/dt)+B, andNu=vu’, wehaveinoperator format 278 Cuarren 12 Lu+Ru+ Nu=g. Decomposition results inu=0~,u,with un=c)+et+L'g or Up=co+et+>eafnenia 2) Ed up=C,+e,t+got7/2-14+ gv/3-2 +g,¢/4-3+ 20° /5-4+- The following components are: u,4,=-L'a(d/dtju, -L"Bu,-Lua, where theA,foru’have been given byconvenient algorithms. Thus, u,=-aet' /2-Bot? /2-ag,’ /6-Bo’/6 ~arg,"/24~Bgnt*/24—as.t° /60 —Bg,t'/120- ag," /120- Bg,t* /360-- ::(ye cm ~re [oie /2—|Tie.Tekye8 4) {xe_vers,|Freee\s(20740 2) We can continue toobtain us.us, ....However. itismost convenient towrite thesolution inascending powers oftas ude where ¢.and¢,aregiven andm =~20 Be_16,Bo nrar ria’ c=FyGite,Ye,BeBo_yeiss_ 6 6 6 6 6 2 6 SoumowoFmeDurrivcEgusow 279 Lo a tye gt 2=-2OBotro,alee,BG|area_aeyr ircaina aT2, A Xs 2BeBrey Boo1°)7658)7G, Be24 6 24 8 8 4 12 °° "720120120120 40 2a7c, o's, afc, aByo, abe, ay", 4 1200:«tCS( (CC PLES) ,OFC OB:,Bey,Bycic,Be..g7"cxcy 20 3 60 10° 5 120 40 40 20 20 "20 cya2Bey_a7,a's,,BG°720-720-720 *720*180 aScic a's, aBley are) a*Be, ,a7'c 60 720° 240 +«60 «©2240 ~—«80 2758 Maryech,a*g, afte, 2aBresc, |abs, 80 120” 360 240 15360 Tara ,areie,13076%8.,Tare;a8,BeSB*rch 48120 120 120-120 720 ~«144 4fi8o IBYcs,TBYC85,IBYyc}Be,3775,7*crg5 720 240 120 120 360 80” 16 TCC _YCGR, _YCGS, 1VCRs_1TB),Se40 2030 40-20-30 Thesum°~,c,t®isthesolution through thet*termandifwehavec,or¢, equal tozero, theresult becomes quite simple. RESPONSE OF NONLINEAR STOCHASTIC OPERATORS: Random vibrations arise, e.g., inspace structures andbuildings subjected to seismic events. Ourobjective willbeconsideration ofrandomness inphysical systems, which aregenerally nonlinear, without theuseofperturbation or linearization which may prevent ourseeing realpossibilities ofcatastrophic failure. We will also consider parameters and excitations without theusual 240 Cuarren12 restrictive assumptions which arecustomary butdonotnecessarily conform to physical reality. Forthepresent, assumetheequation u”+au’+Bu+yf(u)=g orLu+Ru +Nu=gwith L=d?/dt?, R=«a(d/dt)+B,and Nu=yf(u). Wecan consider cases inwhich oneormore ofthe,B,7,g may bestochastic processes, without restriction toonly gbeing stochastic, and without further assuming awhitenoiseexcitation. Decomposition yieldsu=.~, u,with u,=u(0)+1u'(0)+L"'g U5=—L%a(4/at)u,.)- LBu,v, wheretheA,aredetermined forf(u).Theapproximant @,=“ u,,serves asthesolution. Since theprocedure converges rapidly sothatafewterms are sufficient inpractical cases andbecause theterms depend onpreceding terms rather than following terms, avoiding closure problems, ensemble averages can betaken term byterm todetermine <u>.We domake thenatural assumption thattheexcitation andparameter processes areuncorrelated. Then taking theensemble average oftheproduct ¢,,(t) dy(t"),we canalsogetatwo- pointcorrelation. Areviewofthenecessaryknowledgeofstochasticprocesses forapplication tosolution ofphysical problems bydecomposition appears in 0). DUFFING’S EQUATION WITHOUT PERTURBATION TO GIVEN ACCURACY: Quantitative general solutions ofDuffing’s equation areeasily found using thedecomposition method. The motion depends ontheinitial conditions u(0),u’(O),theparameters, andtheinputs,Themethodofsolution makes no assumption onthenatureofthe output oronsmaliness ofcertain parameters, andisnotrestricted toasingle input orcloseness oftheexcitation frequency and thenatural (unforced) frequency. This section will show that byseeking solutions toonly thenecessary accuracy, considerable computation and difficult imegrations are avoidable. The appearance ofharmonics and subharmonics will bedemonstrated. Finally, wewill demonswate, using the Durffing equation, thatdecomposition subsumes perturbation. Souumon oF Te Dureine Equariow 281 u”+@u’+@ju+Bu’ =g(t) LetL=?/at* andL"=f'f"()atdeandsoiveforLu.Thus, Lu=g(t)- @u’-wju-fu’ L'Lu=L"'g-Lau’-L'oju-L" Bw? u=u(0)+w’(0)+L"g-L'au’-L"oju-L" Bu? Letubedecomposed intocomponents J,u,withu,identified as u,=u(0)+tu'(0+L"'g with other components tobedetermined. (The nonlinear term iswritten as Dz,Aa{u’} orbriefly asS",A,.)Thetermsafteruyare: Uge=-L uy-Laju, -L"B A, form>.0. Then ¢,=" u,approximates thesolution u=™, u,ina rapidly converging series. Although this provides general solutions, there are difficulties with trigonometric inputs, forexample, andtheu’term; wecanstillgetdifficult integrations despite defining Lsothat adifficult Green’s function can be avoided. Also wecan getaproliferation ofterms, causing unnecessary computation. Hence, wewillassume g=>”,g,t”which might appear counter-intuitive becauseoftheproliferation problem,However weonlyneed tocompute toanecessary accuracy inaphysical problem which we demonstrate withsomeilluminating examples. EXAMPLE: Consider u”+u’+u+u’ =g(t) where u(0) =2andu(0)=-1 and g=cos’ t+3e™ cos*t+3e™ cost—sint+e™' +e Ifweapproximate each function ingwith theterms ofitsseries through and calculate L*'g, weget917/2 totheabove approximation. Hence 282 Cuarren 12 u,=2-1+(9/2)t? andu,=-(9/2)t*. Therefore thetwo-term approximation, correct through the 1?term, isu=2—t which we can write as us(1-t+1)/2)+(1-17/2)=e' +005t,uj =g=cos’t+-andu=costforlarget_Notethatthisistheexactsolution. EXAMPLE: Show supetharmonics arepossible inaDuffing equation. u’+u’+utu=g=t+10t+- u(0)=1 u(0)=2 Where gisatrigonometric function asbefore. Toavoid difficult integrations, weconsider theMaclaurin approximation tothree terms: ga347+100 Since u,=u(0)+1’(0)+L"'g we have uy=1420~38 ifwedrop terms greater thant°,Then u,=-Lu, -buy -Lu) =2° The two-term approximant 9,isgiven by steak° 2 which weseecanbearranged as¢.=(1-1°/2)-2t andimmediately guess w= cost+sin2t showingexistence ofasuperharmonic. Neediess tosay,weverifythesolution obtained bydirect substitution. (Also wecan consider another term ofthe Maclaurin expansion ofgtogetthecubicterminourapproximant.) EXAMPLE: Show thatsubharmonics canarise inaDuffing equation. SoumiosoFTueDuFFiveEgvaTion 3 u’+u'tu+w=g with u(0) =2andu’(0)=0. gisagain atrigonometric form whose Maclaurin expansion isgiven by =U ue aaT) Then 79bigs2s=8 throughquadratic terms.Thefirsttermofthe decomposition series is uy=u(0)+0u'(0)+Lg2 Solving theDuffing equation bydecomposition, wehave u=u,-L'u’-L"u-L’ Since™, u,,thenexttermu,is uy=—Lu, Lu,-"A,{u"} Thepolynomial A,{u’} issimply up;therefore, u,=5t plus,ofcourse, higher terms. Thetwo-term approximant 9,tothesolution isu,+u,;hence B2_g2-5 1a=2+40-st 22-4 %18 18 which werecognize as a ia =| 1-4 |+]1-i+e. or 1 = cost+cos=t 3 Decomposition yields thesolution—it depends ontheparameters, given conditions, and theexcitation, 284 Charren 12 Thefactthatwerecognize closed forms hereisperhaps interesting butofno real significance. The series canbecarried farenough forcomputation as necessary. Thetraditional emphasis onclosed form solutions hasgenerally led toreplacement ofactual problems with more tractable butlessrealistic models. PERTURBATION VS. DECOMPOSITION: Consider thehomogeneous Duffing equation with nodamping andassuming “small”nonlinearterm:u"+uteu=0 u(0)=a u'(0)=0 Using perturbation define u=u,(t)+eu,(t)+---. Hence, substituting toO(e), uy+eu+u,eu, +(u,+eu,) =0 Equating powers of€, vytu,=0 The linear solution (€=0)satisfying thegiven conditons isu,=acost The e!terms give us ulu,=—u} a"cos’t=-a"(3cost +cos3t) withu,(0)=uj(0)=0. Hencewecansolveforu,andwriteu,+eu,.We observe thattheu,isthesolution ofu+u,=0. Indecomposition itis simply u,=u(0)+tu’(0). Also theperturbative term isharder toobtain than thedecomposition component u,=-L“'u, -€L™'A, where theL”ismerely a double integration. The perturbation case involves integration using aGreen's function andmore difficult integration. Further, theperturbation u,involves thesecular term tsint,andwedonotgetauniformly valid expansion which would allow abounded ufor afinite number of terms. Thus U,/Uy>easte°.Theresultsconverge slowlywhiledecomposition converges rapidly sofew terms arerequired. The totals should bethesame but notterm byterm. Applying decomposition totheequivalent linear system u”+u=0 with Soutmow oFre Durring EpusTion was u(0)=a andu’(0)=0, thedecomposition terms are uy=a u,=-L"u, =at"/2 u,=-Ly, =at"/2 Therefore, thethree-term approximant obtained bydecomposition is eo sa 1-4o=qi-£+5) whichistheapproximant tou=acost.Nowconsider u”+u+Bu’ =0by decomposition without assumption ofsmallness (and useofperturbation). We get u)=a u,=—-Lu, —BL"u} =-a7/2-Ba’e?’/2 u,=-L"u, BL“? ‘Wehave added —BL“'u} or—Ba’t?/2 asafirstapproximation totheprevious linear result. When thesystem isclose tolinear (weakly nonlinear), weget Bee. vu’tuteNu=0 u(0)=a and u’(0)=0 u,=a u,=-L"'uy-eL Nu NowtheeL™'Nu approaches €L”'uor—at?/2. Forthisweaklynonlinear (or small €)case thefirst approximation or€term u/+u,=—acost so u,=-at?/2. Thus thefirst decomposition addition isequal tothefirst-order perturbation result ifandonly ifthenonlinearity Nuissufficiently small. Indecomposition there isnotarestriction to“close tolinear”; itapplies generally tonon- linear systems so“weakly nonlinear” or“linear” become special cases, 286 Charren 12 u,=-Lu, =-at?/2 e . a=(1-5)@costasthenumberoftermsincreases which iseasily checked byfinding more components. Intheequation u”+u+eu’ =0, wehad uy=a “1 ayu,=-L'u)-eL us Forsmal] enough ©,wehave added -L"'u} or-a°t?/2 asafirst approximation tothelinear result u=acos. When Nu=u, i.e., wehave @ weak nonlinearity, weget vu"+useNu=0 u(0)=a and u'(0)=0 usa u,=-L"u, -eL"u, TheeL'Nu or€L”'A, approaches €L"'u, or-at*/2. Intheperturbation case thefirst approximation or€term now satisfies uy+u,=—a cost so u,=—at"/2. Thus thefirstdecomposition addition isequal tothefirst-order perturbation result. Perturbation iseffective ifandonly ifthenonlinearity Nu isalmost linear. Decomposition iseffective forgeneral nonlinearities and includes perturbation asaspecial case. Discontinuities infrequency response will occur asaresult ofvarying excitation frequency, since thenonlinearity acting onthedifference between excitation frequency andnatural frequency causes new frequencies toappear andnew multiple possible responses. With small damping. theoscillatory motion can suddenly change from slow tofastorvice-versa. Inphase space wecanhave changes from oneorbit toanother andmay find separated regions dependent oninitial conditions, parameters, and excitation. Ifwechange excitation frequency toapproach thenatural frequency, thebehavior can change significantly. Decomposition yields theactual quantitative results for Souumow or ne Durrive Eguarion 287 real physical behavior forany given parameters, conditions, and inputs whether constants ortime-varying. However, conditions must bespecified. Further, decomposition provides solutions forreal oscillators with any nonlinearity asdetermined from laboratory measurement, notonly those with a simple nonlinearity u’which might be,inactuality, u% SUGGESTED READING L.A.Blaguiére. NonlinearSystemAnalysis,AcademicPress(1988) 2. J.Hale, Oscillations inNonlinear Systems, McGraw-Hill (1963). 3. C.Hyashi. Nonlinear OscillationsinPhysicalSystems,McGraw-Hill (1963), 4.G.Duffing, Erwungene Schwingungen beiVerdnderlicher Eigenfrequen und ihre technische Bedeutung, Vieweg (1918). 5. 1.Guckenbeimer and P.Holmes, Nonlinear Oscillations. Dynamical Systems. and Bifurcations, Springer-Verlag (1983).6.K.Kreith,Oscillation Theory,Springer-Verlag (1973).7.P.Hagedorn, NonlinearOscillations, 2nded,Clarendon(1988).8. 1.D.Cole, Perturbation Methods inApplied Mathematics, Blaisdell (1968). 9. ALA. Andronov. A. A. Vite and S.E,Khaikin. F.Immirzi. transl. Theory of Oscillators, Addison-Wesley (1966). CHAPTER 13 BOUNDARY-VALUE PROBLEMS WITH CLOSED IRREGULAR CONTOURS OR SURFACES ‘The simulant concept cannow beused inanextremely valuable application, thatofboundary-value problems fordifferential orpartial differential equations modelling physical problems between two closed imegular contours (or surfaces). These areconsidered using decomposition oftheboundary shape andsimulation ofthesolution foreach boundary approximant. Ourobjective istosolve “two-limit" boundary-value problems analogous to two-point boundary-value problems forasecond-order ordinary differential equation with Dirichlet conditions. Intwo dimensions, thecorresponding situation isatwo-contour second-order partial differential equation. Inthree dimensions, the analogue isathree-dimensional second-order partial differential equation solved between twosurfaces. Wecancontinue toann- dimensional second-order partial differential equation and n-dimensional manifolds. Our special interest isinsolving partial differential equations in regions bounded bycontours orsurfaces. TWo-CoNTOUR Cask: Thus forthetwo-point, i.e., two-limit problem forsecond-order ordinary differential equations, which wecan think ofasaone-dimensional partial differential equation, we have two-point boundary conditions u(s)|gag,=b, andu(x)},.¢, =b,wherex=é,andx=,areembedded ina lineandweareconsidering equations such asd*u/dx’ +f(u,u’) =0.Inthe two-dimensional case weconsider equations such as with conditions such as 4yfcaro Bi U(Y) tayo Ds Thus the limits are smooth closed curves orcontours. ans BouwoanY Vauve PROBLEMS Wir CLOSED IaREGULAR CoKTOURS oRSURFACES 289 TWo-SURFACE BOUNDARY-VALUE PROBLEM: Here, weconsider equations such as ut, +U, +p(%y.z)E(u,u,,u,)=0 inathree-dimensional region bounded bysurfaces S,(x,y.2) =0andS(x,y.2) =embedded intheregion. Our boundary conditions are (5,942)00%by U(K,y,2))o, 00%Bs Thesurfaces aresmooth closed surfaces representing thelimits. Obviously the concept canbeextended toequations such as Sau ——__*______=0OK,+RipeX,)E(Ust,) with UCR)fayao= bs U(R)fgcxj0 De where X=(x,,...,%,).M; andM;aresmooth closed manifolds representing the limits inndimensions. Letusbegin with themerely illustrative one-dimensional example for comparison, using decomposition [1].Consider atwo-point simple one- dimensional boundary-value problem d’u/dx? +vu=0 with vanumerical constantandDirichlet conditions u(x=&)=b, andu(x=,)=b.. Thefirstexampledoesnotrequirethetechnique ofanalytic simulation butserves asan introduction tothefollowing multidimensional cases. The equation can be written inthedecomposition form asLu+Ru=0.Solving bythe decomposition technique yields DYu=c+ex-vE>uy, Forsfod 200 Carrer13 where I?isatwo-fold pure integration with respect tox.Bydouble decomposition, Ford a emt Hence yD wr=L Pend vey Yurmm cod Ext ma We now have uy=ef?+xel®? u=cl +xcl vu, u,=e?+xc!vu, u,=)" +xe\-v Eu, Wecanwrite wd ur Fd u=Yu=y Yur a has uy=(vy?{oftx7*/(2n)!+ of"x"!/(2n+1))} uy=ci=x02)+(vy{ele x3/(2n)+effx"/(2n+1)!} Inthedecomposition method the(m+1)-term approximant tothesolution uis symbolized by0,.,=", u,.Thus =U, O,=0,+u, Far =Oy+Ue Bousoare Vauve PROBLEMS WITH CLOSED IRREGULAR CONTOURS ORSvRFACES 201 ‘Theexact boundary conditions u(x=§,)=b, and u(x=,)=b, canbe approximated successively bytheapproximate boundary conditions a(x=S)=b, 9(x=S,)=by Gaui(8=S)=D, Gan(X=5s)=bs Butsince 9,_, =0, +u,.then u(x=51)=b, u,(x=)=b: u(x=G)=0 u(x=)=0 u,(x =) =0 u,(x=3,)=0 Define b{”=b,andb{”=b,.Then, be=D vy"{otSF[anys g"""/(20+ 1)} be)=cv)"{ol/(2nyt cl") 2"!/(2n+1} a LOE ol) = cyt)+,cf)=be? or 1&)(of)_(o™ 1& Jam) (oe Thus for&,#&(a)_gpm) yoy-S,B=8,BP) oeBA got=$1 Using staggered summation 22 Charen 13 w=Sdu=y Yule u=yYu=y Yu Bo me ywr=L Lu a mm Recasting thisexample intheformat ofhigher dimensional cases, @u/dx?+vu=0 UOD|paro™ Pr UR)apo=be where P(x)=x-, and P,(x)=x~E,, Weconsider thesimulants ¢,,to theapproximations oftheboundaries and denote them by¢,[u] which becomes uinthelimitThus (#/dx*)o, +vo, =0 oalhagn =, Fu(%)|gag) =bs Successive simulants are6,,0;....Fq. Thelim(x)= u(x).Intheone dimensional case itisx==p, a“radius”. (We may develop apoint sequence where lim£(*)=&,andsimilarly for4°, Thus limo, =u.)In thetwo-dimensional case, thelimits arenotpoints butclosed contours inR* described byC(x,y) =0andwecanhave acontour sequence C'®’(x.y)=0 Ifthecontours arenot smooth butconsist, forexample, ofpiecewise differentiable functions, wecan represent them bysmooth continuous functions asaccurately aswewish, andwithout Gibbs phenomena, byarecent combination oftechniques fordecomposition ofalgebraic and differential equations [2].Thus wecanassume that thecontours (orsurfaces) aresmooth though irregular inshape Bouvoaky VALUE PROBLEMS WITH CLOSED IRREGULAR CONTOURS ORSURFACES 293 TWO-DIMENSIONAL CASE: Now weconsider atwo-dimensional case withthemodel equation onR* Puldx?+Fuldy=0 which weview asatwo-dimensional analogue ofthefirst example with v(y)=0* /dy*.Analogous boundary conditions are UOy)feyupne =P u(y) eggusyeo =bs where C, and C,areclosed contours representing theboundaries inR®. (We can, ifwewish, lettheouter contour —>=ortheinner contour approach theorigin.) The model equation iswritten asLu+Lju=0 where L,=2°/0x? andL,=0°/dy*. Operating withLy, LiLu=-LiL,u usce(y)+xe,(y)-L, Ru Decomposing winto)", u,wehave D4=co(y)+xe,()- LEYvy Ef Fa Usingdouble decomposition u=~, Y, ul).Also, coly) = Cy) emt aly)= >Cy) a=yuo a DL uP=Y MO)+xd M)-L LY Ywe Bas End Em mm 204 Cuarren13 ug=en"(y)+xe)"(y) 1,=65)(y) +xe}"(y)—L Thup u,=p'(y)+xeP(y)-L, uy, ue=G(y)+xe"(y)-L,Hue, Wecanwrite ued ue s ue=(Ly)felx28/(2n)!+of")x2"/(2n+1)}}ad or ug=e."=xej" +)(-L,fer x7*/(2n)t+ cf"x?"/(2n+»} Theapproximant tothesolution is9,,,=)", u,-Theexact boundary conditions are: U(Xy)Jeyayeo =bs USy)Jestayie0 =Bs areapproximated byO0-1(%-¥)feonsie0 =b 4.210¥)Jeyiagyo =Bs form=0,1,2,....Since@,.,=0,+u,,Wehave UCY)Jesayeo =O U,(%y)feyrarieo =O Ug(%Y)Jojayyo =O 4,(KYegrageo =O Bouwoanr VALUE PRoBLeMs WITH CLOseD InegcuLan CONTOURS ORSURFACES 295 Theinterior contour orboundary isgiven asC,(xy)=0 sothatx=&(y) The exterior boundary C,(x,y)=0 sothat x=é,(y).’ These can be approximated by&;""(y) and &"(y). Ifwewish foraparticular model, wecanconsider lim&{")(y)+©sotheexteriorboundary +=».Define b=, andbY”=b,and &{EO)yyCoen(yy4SY)(L comany)} be=->(YL cen --L,)ce"y2(Gn):(LYero)a lt)(ot ve=-5,[Q)(LJ.ceQ)+ ey)(4,)cen(y)}PAEGayho)MO)Gas) MOD} ‘sowecan write cf+2,cl=n") clo)+2,cle=n or. 1&4) (cy) _(oi") 1&J}lem) Thus fo)=S204 0&-§ cobe ara THREE-DIMENSIONAL CASE: Wecangeneralize toR’withtwoclosed surfaces S,(x,y,z)=0 and S,(x,y,2)=0 (oreven tomanifolds inR*)andsimulate thesolution u(x.y,2) representing themodel phenomena. TheDirichlet conditions are U(Xy.2)fsayaye0=b, fori=1,2 ‘Thesimulants areo.[u]>uforsufficiently high-order m,ic,lim0=u. ‘Theboundary shape issubjected todecomposition andsimulants arefound for ™Seeimplicitfunctiontheorem. Alsoseechapterondecomposition ofalgebraicequations (1). 296 Cnarren13 successive boundary approximations tohigher andhigher order. Thus satisfies themodel equation andtheboundary condition (922) yoo=Bs for i=1,2form= 1,2,3,... ASweincrease thedimensionality, wecanuse theprevious results bywriting Lu+L,u=0 where L,=°/x* andL,.=0*/dy* +d*/2*, Then Lw=-L,u LyLu=L Le where y’=(y,z). Decomposition u=J",u,yields Dw Cy)exC(y)-Ly- ED vs Withdouble decomposition u=~~, u!®and Cly)= YLCry) as Cly)= DdcP) w= uf? DLowa LTcre xqry’)-Ly YD wy sothat uy=Cy’) Cy’) B=Cy)+XCMy)-L,Huy BOUNDARY VALUE PROBLEMS WITH CLOSED IRREGULAR CONTOURS OR SURFACES 27 =CMY) +xCM(y')-Ly Ru, Ug=CMy)+xCMy)-Ly Betas Wecanwrite w= ue =} w=u-FSue Thena7 Oe u,=by(-L,.Y{esx?/(2n)!+Ce"x"/(2n+1)'}Et uy=C+xCl+>(-L,Yio x°*/(2n)t+ Cf"x7/(2n+1)!h Our (m+ 1)term approximants tothesolution are an=)Ys Ef BaUy Or.=O+Uy Past =Om+Uy The exact boundary conditions are usIfcrro =P: UyVscarreo=Pe ‘The approximate conditions are 4a(%Y fearro=P 4_(%Y'Ifstureo =Ps forsuccessive mbutsince ¢,,; =Oa+a» 298 Charen13 ug(%y)}sje0 =bs ua(%YIeje0 =bs form= 0,1,2,3... S,(x,y’)=0 implies x=&(y’) andS,(x,y’)=0 im- plies x=&(y’). Hence x=€'")(y’) and x=E")(y’) aretheapproximate explicitfunctions. Defining 6°=b,andb®)=b,. pee SLE CLten, EE tes‘2oo SVC aay Je vee a(Jie +ot(-2,)'ce * leas oy Qn+r ho sothat CP)+," =bie? CFE Cape? yielding ci)=bsbe57 G bie) be 2) d ifweexclude thetrivial case 2.4Z,.IflimS™(x.y,z)=S(x.y.2). then limo,[u}=u. The ideas used forthesolution bydecomposition ofalgebraic equations can beused toobtain smooth expansions ofpiecewise-differentiable functions and dosowithout Gibbs phenomena. Thus inourconsideration ofirregular contours and surfaces, wecan ifnecessary gofarther and consider non- smooth contours and surfaces aswell. Physically, this means wecan approximate shapes ofartificial devices. Mathematically, itmeans broadening Ofthe class ofnonlinearities, Consider, forexample, thefunction formed onthedomain 0<x<2bytwo simple parabolas, onewith vertex atx=0,y=0andonewith vertex atx=2. y=0.cg., y=xandy=(x-2)*, Writey(x)=P\(x) for0<x< 1and BounDARy VALve PROBLEMS WITH CLOSED IRREGULAR CONTOURS OR SURFACES 299 y(x) =P(x) for1<x-<2.Now y(x) isnon-differentiable at(1,1). Wewill show thatanA,expansion canbecarried out, ie.,theanalyticity requirement canbeweakened. Write [y-P,(x)][y -P,(x)] =0considering P,(x)andP.(x) asroots ofaquadratic equation tobesolved bydecomposition. Ofcourse, the functions need notboth beparabolas, orforthatmatter, other functions canbe used foreither onewith adiscontinuity inthederivative attheintersection. Consider theexample yx) =x Osxsl y(x) = sxc or (y-xy-x)=0 Thedecomposition form ofthisquadratic equation inyisgiven as x Lo Sap =. Say YO)"TyXian) >+") wheretheA,fy}aretheA,polynomials forthefunction y*.Thedecom- position ofyintocomponents resultsin =x aries) 1 2 =—— fy} 20 You"Fey Maly} on determining y=" ,y,Ann-term approximation g,=Y.*'y, is 9,=(x/(1+x))(1+ x/(1+xf+28/(1+ x)+50/(1+ x)+...) ‘The result can bewritten yx)=(x'/(1+ x)k.(x°/(1+9)"*) ifwedefine ky=1andky,)=anKykyo (Ko=k=1,ky=2, ky=5,ky=14,ky=42,... .)This result now represents thefunction of interest;thus,itisanalogous toaFourierexpansion, Notethatthelimitofyas 300 Cunrren13 x&isx,i.e.,intheregion]<x<ex;thelimitofyasx0isx’,In[1]it isshown that the smallest root ofaquadratic(orhigherdegreepolynomial) is obtained first and weseethesmallest root values in0<x<Jarefrom they= x?section, while intheI<x<e=region, thesmallest rootvalues arefrom y= x.The farther apart theroots are,thefaster theconvergence willbe. Thesecond rootofy*~(x+x*)y+x’ =0isobtained bydividing thisby theroot already obtained, e.g., ifweconsider only @,aone-term approximant ofthefirstrootor¢,=x?/(1+x), wehave fy—(x+x*)y+ x}/fy- xU(1+x)}=0 Thus, neglecting theremainder, thesecond rootisy=(x+x")- x*/(1+x) whose limit forsmall xisxandwhose limit forlarge xisx*,Ifthefirstrootis £,(x),thesecond rootis[v=(x+x3)y+ xJy- £,(x)]or ye(sex-t)+(xe-8 (x+x-§))/y-F, which issolvable bydecomposition. Wecannow donumerical calculations to seehow well theresult approximates thefunction ofinterest. Wechoose one value ofxineach region andoneatthediscontinuity. Atx=1/2,wemust havey=1/4,i.e,limo,=X.Y,givesexactlythosevalues. x=12 x=2 x=l Qi= -1666667 p,=13333333 = 5000000 @,= .2037037 9:=1.6296296 = 6250000 = .2201646 = 1.7613169 = 6875000 9.= 2293096 Gu=18344764 @.= .7265625 Q= .2349998 G=1.879998 = .7539063 Q.= .2387932 g.= 19103456 Q= .7744141 r= 2414426 G=1.9315408 = 79052174 = 2433561 @=1.94682485 = 8036194 Q= 2447734 9,=1.0591975 =9145204 p= -2458464 p= 1.9667548 Gyo= 8238030 aime. 14 dim. =2 ding.=1 1.6% error byPio 1.6% error byPio Bouwoany VALUE PROBLEMS WITH CLOSED IRREGULAR CoWTOURS ORSURFACES sor Wenote from observation ofthevalues atx=|thattheexpansion intheA, polynomials does notdisplay theGibbs phenomenon which weseeinFourier series (2]. Instead, wegetablending effect atthepoint ofdiscontinuity inthe derivative—an interesting application ofthedecomposition method. REFERENCES 1. G.Adomian, Nonlinear Stochastic Operator Equations, Academic Press (1986). 2. G.Adomian and R.Rach, Smooth Polynomial Expansions ofPiecewise-differentiable Functions, Appl. Math, Lett. 2,(377-79) (1988), CHAPTER 14 APPLICATIONS IN PHYSICS Real problems ofphysics aregenerally nonlinear and often stochastic as well. Linearity anddeterminism should beviewed asspecial cases only. The general practices oflinearization, perturbation, white noise, and quasi- monochromatic approximations necessarily change the problems whose solutions aredesired, tobetractable byconvenient mathematics. They arenot then identical tothephysical solutions which weseek. Thealternative ofusing thedecomposition method will beexplored here asweconsider examples of problems ofphysics. These problems areoften quite difficult because nonlinearity andstochasticity areinvolved. Decomposition makes unnecessary procedures such asclosure approximations [1]andperturbation andwhite noise processes indifferential equations which involve stochastic process parameters, inputs, orinitial/boundary conditions. Decomposition alsoavoids discretization andconsequent intensive computer calculation andyields analytic expressions rather than tables ofnumbers. Thus quantitative solutions are obtained fordynamical systems. (When thesystems arestochastic aswell, the decomposition series involves stochastic terms from which statistics canbe calculated.) The method applies tolinear ornonlinear, ordinary orpartial differential equations and isuseful formany algebraic, integral, anddelay- differential equations [2].This chapter will outline procedures fortypical applications ANALYTICAL SOLUTION OFTHE NAVIER-STOKES EQUATIONS: ‘The Navier-Stokes model” 1,2]foranincompressible fluid ofkinematic viscosity v,andconstant density pisgiven as (aB/at)+(B-VG-vV'T+(I/p)Vp=F ueQx(0.T) Vv-a=0 2Qx(0.T) T=0 42x(0,T) where @isavector with components u,v,Ww. ”This weatment differs from theearlier work presentedin[2]inthatpressureisdynamic aliowing farlarge velocity and turbulence 302 Apmuscarions ivParsics 303 Weassume thatthevelocity U(x,y,z,t,@), thepressure, p(x,y,2,t,a), and theexternal force, Farestochastic processes. Interms ofvelocity components u,vow, we write (du/at)=u(du/ax)+ v(du/ay)+w(du/aw) -v{(Fu/ax’) +(#u/ay*)+(#u/az")} wD +(Wp\ab/ax)=F, withsimilar equations forv(replacing F,byF,anddp/dx byp/dy) and forw(replacing F,byF,anddp/dybydp/dz).Wedefineaninitial- boundary problem byspecifying initial conditions foru,v,wandfort>0, specifying u,v,w,ontheboundary. Let's rewrite thesystem (1)intheequation above inthedecomposition form. Lu+N,(uvw)= 3, Ly+N,(u,v.¥)= gy Lw+N,(uyw)=2, We have some choice onthedefinition ofnonlinear terms. Let's consider L=(a/at)- v(a*/ax*)- o(a?/ay*)-v(a*/a2*) =L,+L,+L,+L, N,=u(du/dx) +v(du/dy)+w(du/dz) N,=v(av/dy) +w(dv/dz) +u(av/ax) N,=w(dw/dz) +u(aw/ax) +(aw/dy) 8)=F,~(I/p\(ap/ax) 82=F,-(1/p9p/2y) 8=F,—(/p)(9p/42) Tocomplete thespecification ofg,,g2,g)Wemust know thepressure function, Wecan assume aninitial pressure which will, ofcourse, become a function ofx,y,2,tasanydisturbance occurs. However, wemust determine thefunctional dependence ofpressure onthevelocities u,v,wsothat theg,, 2,8) arecalculable, 404 Cuarren 14 Ifwefind thedivergence ofeach term intheNavier-Stokes equation, the Vpbecomes V’por1/pV’p depending onthedefinition used forp.Thefirst andthirdtermsvanishftomthedivergence condition. Thesecondtermgives usV(U-V)-@. Thus Vip=V-F-V(G-V)-0 Thus, 2 2 2 p_(auy)_(avy _(aw L,+L,+L,}p=V-F-| <2}-|2)-| beet +hp (5)(5)(=) 29dv,dwdu_,dwavdyax dxdz dydz Symbolizing therightsidebyf,solvingforL,pandinverting theoperatorL,, we have p=A+Bx+L;f-LjL,p-LjL,p Writing p=xp,andidentifying p= A+Bx+Lit wehave forn>0 ; Pye =-LiLyp, —LyLp, andweCanwriteann-term approximation forpby¢,=... p,whichcon- vergesto.” p,orp.(Similarequations canbewrittenforL,pandL,p.) Weassumed aninitial pressure which gives ustheAinourequation forpo. The coefficient Biszero since thedisturbance vanishes asx—>s».We use this ptofind u,v,w.The resulting velocities areused inourequation forp,asa function ofvelocities, toyield animproved p=py+p;(which re-calculates py because ofthechange inf).This isused toimprove results forvelocities u,v, w.These calculations canproceed until wehave sufficiently accurate results foru,v, w,p.Wehave Lu+Lu+Lu+L,u=g,-N, Ly+L,v+L,v+L,v=g,-N, Lw+L,w+L,w+L,w=g,-N, from which Apmucaions wvPusics 405 Lu=g,-N,-Lwu-Lju-Lu Lu=g,-N,-Lu-Lu-Lu Lyu=g,-N,-Lyu-Lyu-L,u Liw=g,-N,-Lwu-Lu-Lu Four similar equations canbewritten forvwith g,and N,replaced byg;and N;andalso four equations forwusing g,andNy Ithas been shown inChapter 3thatwhen theboundary conditions aregen- eral(when conditions onanyonevariable depend upon alltheothers) thatto solve foru,v,w,wecanuseanyofthefouroperator equations depending on. thegiven conditions andintegrations required, Ifweknow initial conditions, theequations involving theoperator L,on theleftside will besimplest since only asingle integration will berequired. Wecanalso solve thesystem asa boundary-value problem using anyoftheequations involving L,,L,,orL,on theleftside asdiscussed inChapter 4.Hence, using thefirstequation ofeach ‘setabove andoperating withL;',wehave u=u(0)+Ly'g, -LiN,-Ly(L, +L,+L,)u v=v(0)+Li'g) -L;'N, -Ly(L, +L,+L,)v w=w(0)+L;'g,-L'N,-L(L, +L,+L) Nowwritethedecompositons u=)'", uy.v=D, Vee=ygWe Also, write Ny,Nz,Nyinterms oftheA,polynomials andfinally identify: uy=u(0)+L;'g, . vo=v(0)+Lie, w= (0) +L, Theremaining components ofu,v,wforn>0cannowbedetermined by=“LVA{N JL(L,+L,+L,)uy Van=—LiA.{N}-Li(L, +L,+Ly)ve Wey=HLIAL{N,}-L3(L, +L,+L,)w, wherethenotation A,{+}referstotheA,forthequantityinbrackets.Wenow 306 curren 14 have acompletely calculable system Which, ifweignore stochasticity forthe moment, andapproximate by oad a, eS, wehave found u.v,wton-term approximations, Inthestochastic case, theexpressions foru,v,warestochastic series, i.e., series containing stochastic processes which wemust solve forfirst- and sec- ‘ond-order statistics, where each velocity component isreplaced byasum ofa deterministic component velocity andastochastic component. Wedothisnow inspite ofthefact that theequation obtained byreplacing velocities with stochastic processes may notbethecorrect stochastic model since theequation ‘wasderived deterministically, Anexample ofthisistheproblem ofwave prop agation inarandom medium where itisincorrect tosimply replace thevelocity inthed’Alembertian operator with astochastic quantity; i.e.. «stochastic model must bederived which has the deterministic model asalimut rather than using thedeterministic mode! toobtain 2stochastic model [3].Thus, wemust obtain: (u)=(up)+(us)~(us)=(=(so)+(s)=(6:)-—(1)=(4)+(8)+(a) + remembering that g;,g:,g5 arestochastic, since Fand parestochastic and the A,arestochastic. The two-point correlation foreach velocity component u,v,wisobtained byaveraging theproduct ofseries forthevelocity component attwo space- time points. Ifweconsider, forexample, fixed space position and time scales such thatstauonarity canbeassumed, theergodic hypothesis may hold sothat ensemble averages can bereplaced with time averages ofobservations. Sincenonlinear termscancontain bothfunctions ofasinglevariable, suchas f(u), and also functions ofvariables, such asf(u,v) and f(u,v.w), wehave listed these generalized A,inChapter 3,WecanusetheA,{f(u,v)} since N,. No,Ns can beconsidered term byterm. Now thecoefficients inthe APPLICATIONS IwPaYsics 307 generalized algorithm forf(u,v) will involve three quantities, thederivative is f*! andthesummation isover 11,v.Using theresulting generalized A,,we obtain ageneral solution, Smooth solutions (totheincompressible problem under consideration) doexist forshort times andarecontinuously dependent onthe initial data. Abasic question iswhether theNavier-Stokes equations areanadequate model forreal turbulent fluids. The linear constitutive law used inthederiva- tion means thatderivatives ofthevelocity components u,v,Warenecessarily small, Secondly, stochasticity cannot beconsidered asanafterthought; itmust beconsidered inthe initial modelling. Amore general model due to ‘Ladyzhenskaya haspartially addressed thisissue byallowing nonlinearity in theconstitutive lawwhich leads toaglobal uniqueness fornonstationary three- dimensional flow. Atruly nonlinear stochastic model coupled with thedecom- position method ofsolution may resolve remaining difficulties. SOME THOUGHTS ON THE ONSET OF TURBULENCE: Consider firstavery simple equation whose solution istrivial. Thus consider dy/dx =(y-1) which obviously issatisfied byy=1.Nowconsider theef- fect ofa1%changeinaparameter bywritingdy/dx=(y~1)*+.01.**This nowyields aperiodic solution y=1+0.1tan(x/10) which hasvertical asymp- totesat(2k+1)S1,K=0,41,£2,.... Now, let’s make a1%change intheinitial condition, ory(0) =1.01. We now have ahyperbola y=1-1/(x—100)*** andonly onevertical asymptote atx=100.Thus theeffect inanonlinear equation ofeven very small changes ininputs orparameters canresultinlargeeffectsonthesolution, ‘Suppose now thatvery small fluctuations arepresent intheinput andpa- rameter because ofsmall inherent randomness. Then thesolution could change randomly between thepossibilities above andappears very complex indeed. Now considering theNavier-Stokes system with itsnonlinear terms where there could besmall fluctuations indensity, pressure, viscosity, andvelocities, itisclear thatwecanexpect similar effects anda“chaotic-looking” orturbulent case. The nonlinear terms cause small fluctuations tobecome large fluctuations Webave considered y’=(y—1)+a with a>0.Ifa<0, solution varies between two horizontal asymptotes with inflection point at(0,1). The asymptotes coincide ifa=0. ‘The solution y=|isasingular solution notderivable from thegeneral solution, “*"Ify=0.99,theasymptote movestox=—100. 308 Cnarren 14 while friction terms tend toremove differences invelocities. The Reynolds number isameasure oftheratio ofnonlinear terms tofrictional terms, soitis reasonable that ifthenumber becomes large, thetendency toturbulence in- creases. However, factors such assmoothness ofboundaries and themagni- tude ofinitial fluctuations alsoinfluence theresulting flow. Inthesimple deterministic case, consider onenonlinear term udu/@x di- vided byamolecular friction term vd"u/dx’. Ifuanddu/dx areassumed to beoftheorderU,andLisatypicaldistanceoverwhichthevelocityvaries byU,theratioisoftheorder(U*/L)/(v- U*/L)=U-L/vortheReynolds number. Inthegeneral case, ifwehave afluctuation invorvwecanseethat large changes canoccur inthetendency toturbulent behavior. The best way, apparently, todetermine when turbulence starts istosolve the stochastic Navier-Stokes system aswehave outlined andstudy thebehavior as afunction oftheparameters oftheflow. Acomparison ofadeterministic solu- tionandzstochastic solution withvarying conditions shouldilluminate the problem oftheonset ofturbulence. Suppose weconsider flow inaflatchan- nelasanidealization ofapipeinaplane,Wehave x ¢LLLLLLLLLLL. -— Replacexbyx/£t0makethehalf-width unityandassumedp/dx=0.Write Lu= (a?/dx")u- u(a/x)u u=Lio(F/ax)S u,-L YA, where the A.polynomials aregenerated forthenonlinear term. Then u,=u(0)+tu’(0) u,,,=Li'v(#/ax")u, -LTA, AprUCATIONS lvPa¥sics 409 forn>0. Ifvisconstantanduyisdeterministic, uisdeterministic. Ifu,has random component, thiscomponent willcause new terms tokeep appearing because oftheexpressions ontheright side oftheequation foru,,;forany n> 0,especially from theterm involving A.This isobvious byinspection ofthe A,forincreasing n.Theeffect ofphysically unrealistic change inthesolution byalinearization isalsoclear.Consequently, asaresultofanyrandomness andthenonlinearity, theflow isradically altered— theeffect increasing asthe fluctuation becomes larger. Random boundary conditions resulting from roughness inthewalls will have thesame effect. ‘The general problem may have random initial/boundary conditions. 9is ‘generally taken asaconstant andsetequal tounity; however, compressibility becomes afactor with increasing depth andpmay notonlybeafunctionofz but random inturbulent conditions. THE VAN DER POL EQUATION: u"+au'+Butyu'y =g(t) u(0) =cy y(0)=¢, Let L=d?/at? L'Q= [fora a(t)=) eat* Lu=g(t)- a(d/dt)u- Bu-vu'u* usu, —L'a(d/dtju-L"Bu-L'vu'u with uy=u(0)+u'(0)+L“g(t) =u(0)+w’(0)+ ae?fo+1)(n+2) cod Letu=>uy,wur=7, A,.TheA,canbefoundfrom Aged DYtaal 310 Chapren 14 Since wehave us,allthefollowing components aredetermined from uy=Lau, -LBu, -Ly A, Theapproximants tothesolution aregiven by 9,=Up Gan =bq+Us lim@,=u TheA,areAg=UpuG A,=u'u}+2usugu, A,=uyu;+2ufu,u, +ufu?+2ueu,u, A,=ujui +2usu,u, +ufu?+2uyu,u,, 2uguju; +2u;upu, A,=utu) +2u(u,u, +usu? +2usu,u; +2ujuyu, +2ufuyus +usu;+2u;u,u, +2ujusus A,=ulu; +2ujuju, +usu;+2usuou, 2uiuju, +2uiu,u, +ufu;+2ujuu, +2uju,u, +2uju,u, +2u{uu, +2usuou, Ag=ulu; +2ufuyu, +usu;+2ujupu, +2uiu,u, +2ujugu, +uly;+2usu.u, 42u‘ugu, +2u/u,u, +2u;u,u, +2u‘u.us tuju; +2uju,u, +2u;u,u, +2uzu,u, NOTE: This isdone bywriting u’u*=Nu=N,-N,, writing ))B,foru’ and SC, foru’andconsidering thepossible products, e.g A,=B.C.~B,C,+..+B,C,+B,Co. (SeeChapter2.) Given theA,,theuecan becalculated andrearranged inascending powers oft(see Appendix II)togetsolutions toany required answer. The same procedure aswiththeDuffing equation canbeusedtowriteu=)", ¢,t° andcalculate thec,.However, wecangetaquick approximation asinthe following example. ApruscsrionsiwPaysics, a EXAMPLE: u’+u'+u+u'u? =—sint—(sint)(cos*t) with u(0)=1andu’(0)= 0.Approximating sintbytandcostby1—t?/2, we findL"gdoesn’t contribute, u,=1, u,=-t?/2, so¢,=1-07/2 which we recognize asatwo-term approximant ofu=costandwecanverifyby showing thatitsatisfies theequation andthegiven conditions. BURGER’S EQUATION: The equation isU,+uu, =vu,, forx>0andt20 where necessary condi- tions must, ofcourse, begiven. Wewrite Lu=vLu-uuy, U(r0)=£(x) withL,=/atandL,,=4?/ax*, Operating withL!=f'()dt,wehave u=u(=0)+L/v LuLu, Me Weidentify u(tx0)=f(x)astheustermofthedecomposition u=S)",u, andwritethenonlinearity uu,as)”, A,where Ag=Unt A,=Upu; +ujuy A,=uu, tuyuy +u,uy Ay=ujuy +uzul +uu;+uyu; A= ujuy tu, uy+.tuus, +uu Now thecomponents after uparegiven by uy=Lo Lau. “LA, andwecanwrite them-term approximant which converges rapidly tothecor- rectsolution 9,=J.”u,.Sinceeitherofthepossible operator equations for LuandLu canyield thesolution inthegeneral case where theconditions for t=0depend onxandtheconditions onxdepend ont,itisnolonger nec- 312 Cuarren 14 essary touseboth operator equations asinearlier work. (When theconditions arenotgeneral inthisway, wehave asymptotic equality.) Integrations fora difficult f(x)canbemade trivial bywriting f(x)inseries form andcarrying a limitednumberofterms.IfweusetheL,,uequation, uy=A+BxwheretheA,B areevaluated from theboundary conditions andwenotethatL;repre- sentsatwo-foldindefinite integration, Ifwehaveanon-zero u(x,t=0)=f(x),theproblemissimplysolved.Iff(x)=0,wemustusetheLuequation, KURAMOTO-SIVASHINSKY EQUATION: ‘TheK-S equation isgiven as UF UU, +My +tn =O Intheoperatornotationofthedecomposition method,thisis Lu+Lu+Ru+Nu=0 where L,= a/at L,= va‘/ax* Rus pau/ax? Nu=u(a/dx)u This equation describes problems influid motion, fluctuations intheposition ofaflame front andoscillating chemical reactions. There areanumber of possibilities dependent onthestated conditions. Suppose weknow thatu(x.0) =f(x)explicitly. Then wewrite Lu=-Ru-L,u-Nu anddefine Lj=f(dt.Now Li)Le=-L7'u(a/ax")u-L7! v(a*/ax‘)u-L'u(d/ax)u Substituting u=D7,u,,Nu=0",A,wheretheA,aregenerated for u(9/@x)uoruu’.Thesearefoundas Apruicarions iPasics a3 Ay=Uyus A,=uyuy +u,us Ay=uguy +u,uy +uu, Aysuyul +uu, +tuyuy Inthedecomposition ofuintoJ™,u,,weidentifyuy=f(x).Thenfrom u=f(x)-Li'w(a?/ax) >u,-Li'(a'/ax*) Su,-LDA,B Ft & we have uy=f(x) u,.,=-Li'u(d?/ax? Ju,-Li(o*/ax* )u,-LA, forn>0soallcomponents arecalculable. Wecompute ¢,=") u,asann- termapproximant tothesolution u=).~,u,.Theresults aresufficient fora complete solution iff(x)iscontinuous andn-times differentiable ormay beap- propriately transformed byFourier series. Now g,must satisfy theequation tonthapproximation andexactly asn=, Ofcourse, anumerical result de- pendsonanexplicitf(x). Given boundary conditions onx,such as: u(x=&,t)=b, u(x=,,t)=b, u(x=é,,t)=b, u(x=g,,t)=d, wecanalso solve Lyu which will require four-fold (indefinite) integrations. Then 2 Li0=> extort/atervyfffooas dxdxdx (Ifvisafunction ofx,itmust,ofcourse, beinsidetheintegration.) Wewrite a1 Canter 14 U=Co(t)#C,(t)x +e,(t)x?/2+e,(t)x?/6, -Hi(/vXa/anu-1(V/v)Ru-1k(1/v)Nu whereI,(-)=Oa.Bydecomposition usuy-I(yvya/ayy, uy, aam{sa'forS »,naa Now Up=Cp+O,X— C57/24Ojx° 16 uy.)STL v9/atu,=1{(U/v\(ua/ax?Ju,}-ELA, Since thisisanonlinear equation, wemust evaluate thecoefficients foreach approximant ¢,foreachm=1,2....Alternatively, wecanusedoubledecomposition. Inthiscasewewrite w=du? Fa = uy Aad Am e(t}=>c(t)i=0,1.2,3 ‘Theni u,=ul”=>erofs udeonisa¥N/20404 a “TUv(ue8x?)ug,UIVAS APPLICATIONS twPa¥sics as Matching the solution approximants atthe boundaries determines the components oftheintegration constants asdiscussed inChapters 3and4. THE LANE-EMDEN EQUATION: This isone ofthebasic equations inthetheory ofstellar structure in astrophysics and was recently solved byN.T. Shawagfeh [4]using the decomposition method. Itisgiven by 2 eDer-ate=0dre wheremisparticularly ofinterestintherangefrom0to5withtheconditions T(0)=1 and [dT/dr],,,=0 ‘What wasneeded wasanapproximation which didnotrequire 2tobesmall, Such anon-perturbative solution follows. First, the dependent and independent variables aretransformed” using @=€Tand€=4'"r toobtain a)FO __gumgewe € 6(0)=0 dé which iswriten as Le=-g'"6" where L=d?/d&? andL"isatwo-fold integration withrespect to§.Now =6,-L"g""6" where @=&(0)+§d0/d5|,..=&. Thenonlinearity is£(6)=0" =~, A, where *Thisstepiseliminated andresults generalized inwork tobepublished. 316 Curren 14 Ay=£(6.) A,=8,(6,) Az=836(0)+8;/21"(6.) Ay=5(B.)+8,036(8)+6;/31E°(6.) Now@=~, 8,where =F a=LIA, forn>0.@cannowbewrittenasaseriesintheform O=E+oFH0,Eo +o where thec’saredetermined asfollows: o=-¥3! c,=m/s! «=(7h)eee3 Finally, since @=2T, T=c,é?+c,2*+c,é*+--. Anaccurate andeasily computed solution isobtained with seven terms. NONLINEAR TRANSPORT INMOVING FLUIDS: ‘Anew approach totime-dependent spread ofcontaminants inmoving fluids isprovided bydecomposition which iseasily extended tononlinear and stochastic partial differential equations aswell. First weconsider ‘The one-dimensional advection equation: dul dt>adu/ax=0 O<tsT, OSx<1, a>0 u(x,0)=f(x) u(0,t)= g(t) Bydecomposition andusing thepartial solution fort,wehave Aprucarions wPavsics a7 u=u(x,0)-L;' a(a/ax) Yu, where “ L,=d/at Lu=f,(dt u=d uy, u(x,0)=f(x) isidentified asu,,andf(x)isassumed differentiable asnecessary. Then Up=f(x) u,=-@ L;'(a/Ax) f(x) =-aef’(x) u,=(at?/2!)£"(x) uy=($1)*(a*t?/n!) f(x) ‘sothat uD(-1)°(a*?/ni)f(x) andm Gans=D,(D(a? /nt)f(x) @) = isan(m+1)-term approximation tou,satisfying the equation andthe condition att=0using thet-dimension “partial solution”. ‘The x-dimension partial solution isderived by aduldx=-du/at Lu=-a"(a/at)> u, = u=u(0,t)-a L3(a/ay)> u, Fd Consequently, a8 Chaoren 14 up=a(t) u,=a L3(0/2t)u =~"x(0) uy=a7(x7/220): a) Gon, DEa*(x?/21)2() a Either thetequation orthexequation represent thesolution under general conditions. ADVECTION-DIFFUSION EQUATION: Let&(x.y,z,t) represent concentration. Letthefluid velocity beuwith components u,v,winR’andassume anincompressible fluid af/ateu-VE=DVE whereDisthediffusion constant (which isaconstant foraparticular fluidor contaminant, temperature andpressure), &(X.y.2,0) isagiven initial condition and various boundary conditions arepossible. ¢.g., $+0asx,y,z. or &(t)isspecified onaboundary I’,or,wehave apreassigned fluxat. We have 05/at=D{a bax?+aE/ay*+Fé/a2"} ’“ wat /ax—vae/ay— wAF/az Bydecomposition ,usingL,=/dtandL"’=f(ar E=E(t=0)+DL;(a'/ax") DE,-DL;{a*/ay’) D&. +DL}'(8 /4z") YE,-Li'u(a/ax) Y&, -Liv(8/8y) D&.~Li'w(a/ez) D&, 5)=E(t =0)=flx,y.2) Sea =DL VG Lit Vee Appucarions iwPavsics 319 form>0.Now allcomponents are determined and we can write os(€)=ae &,,a8anapproximation to€,improving asNincreases. Ifwehaveturbulent motion ofthe fluid, wecan have random fluctuations of theconcentration andhydrodynamical variables; hence statistical description becomes necessary. ,()becomes aseriesofstochastic termsandweform (0,(€)) togettheexpectation asafunction ofaverage velocity components. The customary treatments ofturbulent motion lead toalack ofclosure and concomitant assumptions which areavoided byusing decomposition. Thus, in theabovemethod, u,v,w,and§arereplaced bytheircorresponding steady-state values plus quantities representing fluctuations from thesteady-states. Thus £=€+8', u=u+u’, vav+v’, w=W+w’, Statistical averaging causes terms such as Dvg) (a/aryé') W(a/ax\e) (wY(/axjs etc. tovanish. We then have (a/NE+B(a/ax}E +Va/ay)E +W(a/az)— =DV*E-w(alaxe -Vdlayye -wajane The last three correlation terms involve correlations ofvelocities and concen- tration which areunknown, Then theprocedure istoletu;fori=1,2,3de- note u,v,w, andx;forj=1,2,3represent x,y,z andtowrite terms asbeing proportional toamean gradient oftheconcentration interms ofa“turbulent diffusion tensor” -K,(x,,t)@E/ 9x;.Toclarifythedifficulty, consider the operator format Lu+Ru=g oru=L"'g-L™ Ru.Ifweaverage wehave (u)=L“'(g)-L"(Ru). Wecanthink ofgasaninput toasystem containing R.Theoutput ucanbestatistically independent ofgbutnotofR.Toachieve closure, onemust approximate. Bydecomposition onewrites usL'g-LR Yu, =L'g-LRL'g+LRLURL'g Et Averaging isnoproblemsincegisstatistically independent ofR.Wehave 320 Cuarren 14 (u)=L'(g)- L"(RYL"(g) +(RL"R)(8)—--- NONLINEAR TRANSPORT: Let's consider theequation LE+RE+NE=gwhere L=a/a NE=£(6) R=u-V-DV" Let=~, &andNE=~, A,.Then geLie-E RYE-LDA, where So=Lig San=LR, LIA, form20.Then ,..=>.., 6which converges to$=)", é,.Further generalizations arestraightforward. We can, for example, consider Fu=¢ where FusLu+Lju+L.u+Lu+Ru+Nu=g andsolve forL,u, L,u, L,u, orL,uwhich would simply treat theother op- erator terms astheremainder operator Randwould require theappropriate given boundary conditions. The case ofstochastic gorstochastic processes intheRterm leads toa stochastic 0;which canbeaveraged orfrom which expectations andcovari- ances canbefound. The solutions areverifiable bychecking that theoriginal equation andthegiven conditions aresatisfied, Since theconcern here issolution ofphysical systems, inputs andconditions areassumed tobebounded. Ifthemodel equation andtheconditions are physically correct andconsistent, asolution isobtained which isunique and accurate. Ifnumerical results arecalculated, onesees theapproach toastable solution forthedesired number ofdecimal places. Ifconditions ononevari- able arebetter known than theothers, weconsider thecorresponding equation which canyield thesolution most accurately. APPLICATIONS ivParsics 121 THE KDV EQuaTion: Theequation isu,+au, +Bu,,, =0. Indecomposition form, wewrite Lu=-awy, -Buy, where u(x.t =0)=f(x)isgiven. This equation was previously solved [2]also using theoperator equation L.u=-8"u, -of"'wu, where L,=4°/2x°. This wasdone toensure useofalltheappropriate ini- tial/boundary conditions. However, inChapter 3wesaw thatthesolutions from each oftheoperator equations—called “partial solutions" —are actually thesolution and identical inthegeneral case when thetconditions depend on X,asabove, andthexconditions depend ont.Therefore, wecansimply use one ortheother saving considerable computation. Since solving theLuequa- tion involves asingle integration andtheL,uequation involves three integra- tions, theoptimal procedure isclear. Weproceed byapplication oftheL;' which isasimple definite integration from 0totThus u=u(t=0)-aLj'u, -BL;'(0/dx’)u Identifying u(t=0)asuyinthedecomposition u=J,u,andwritingthe nonlinear termwu,asJ",A,{uu,}wehave: u=u,-aL;' }A,-BL(3/ax’)> u,co fd ifoandBiareconstants, Wecannowwritethedecomposition components uy=f(x) u,=-aL'A,- BL(9/ax’uy u,=-a@LA,-BL3(2/ax)u, uy=-aLIA, -BL(P/dx us, Indicating u,asu’,wecanwritetheA,forwu’as: 322 Cuapren 14 Ag=uguy A,=ugu +U,u5 A,=ugus +uu;+uzuy A,=ugh, +++ uu, Wecannowwritethem-term approximant tothesolution as6,==)u, which converges tou. THE NONLINEAR KLEIN-GORDON EQUATION: u,—Veu-f(u)=0 Now L,=4°/0t? andL;'isatwo-fold definite integration from 0totandwe canwrite Lu=V'u+f(u) u=A+Bt+L) Vu+Li'f(u) Then. uy=A+Be =L'V'uy +Li'Ag{t(u)} u,=LV, +L'A,{f(u)} uy=L'Viug +LAL ff(u)} ‘Thus, with specification oftheexplicit function f(u)andtheconditions ontas- suming dependence onx,y,z,wecan calculate ¢,,.Wecan useone ofthe otherpossibleoperatorequations iftheappropriate conditions onx.y.orzare better known anddonotgive avanishing upterm. Ifaforcing function gis alsopresent, theu,term becomes uy=A+Bt+L;'g NONLINEAR HEAT EQUATION: cou, =[k(u)u,), With L,=4/At andL;asintegrations from 0tot Avmucarions ovusics $23 1aLu=@Sylk(uN(0/9x)4] u=u(t=0)+ oeZing] Weneed toknow u(t=0)which weidentify asuyinthedecomposition, and also need thefunction k(u) forthespecific mode! being studied. For illustra- tion, weusek(u) =1+uwhich gives usanonlinear term uu,oruu’. Now Lifes ac nl usutoule wetFSMelai Ay=Ugus A,=ugus +u,u5 A,sup +u,u; +uuy A,=ugu ++ u,us Now 1of2 au,-su[Ze -Za,] Life 2 1)=—L;'] Su, +2,ar[OOK| and6,=7 Ua. RANDOM NONLINEAR HEAT EQUATION: WecanwriteusingL,=/3t, a du Lue] KuytweZfuoe]ae and assuming K(u)=1+o(,o)u a au Lu=S}(1+auwZfo-an]es 324 Cuapren 14 uy=u(x,0)+L;'g 4=UFy +2a,2Oe OK Pax Foreither random gorrandom @,forstationary ornonstationary cases, <u> isfound bywriting outtheseries andtaking term byterm expectations without closure approximation orperturbation, Ifgisrandom, <u, >=u(x,0)+Lj' <g> <u,oe. settZcar<y, >eu> nL So>+L5 a>5<Ue where wenote that theinput ofgwould notbecorrelated with a,which isa system parameter. Ifa:israndom, ae a2 a\ <u>=Li Su +L'2|<a><u>2u, SINE-GORDON EQUATION: Fu/dtax=sinu Letting L,=4/AtandL.u=u, =u’, Lu’ssinu v’=v'(0)+L} sinu=u(0)+Ly >A,{sin u} u(0)=u, ifw= uy Fd up=Ju'(O)dx =u(0) uy=LyA, u=[(LFAy)ax= LyLIA Thus thesystem iscalculable using theA,forsinu: Apmucarions(v Pasics ns A,=sin Uy A,=u, cosUy A,=~(u?/2)sin uy+u,cosuy A,=~(u2/6)c0s uy~uySinuy+u,c0sus SCHRODINGER EQUATION WITH QUARTIC POTENTIAL: Mey aye2mox?2" Ey describes thecaseofaparticle inaquartic potential V(x) =(I/2)ax*. Itcanbe written @y/dx* +ax*y+By=0 IfL=d'/dx*, L=[J()axdx, Ly+ax‘y+By=0 yv=O-L'(ax*+A)y Thus: W= where ®satisfies theconditions, Then forn>0 Vou=L'(ax* +B)y, v=DZ) vsisthesolution, ALINEAR EXAMPLE: @u/dx?-keu=g u(t)=u(-1)=0 Wehave, using decomposition, Lu=g+kx’u 326 Cuarren 14 Assume gisaconstant. ThenU=c,+¢,x+gx?/2+L"' kx’u isdecomposed into ug=¢,+,x+gx?/2 ug,=ku,—(m20) Sinceu=7,Uys usby(L'’)Pu,, usby(LiPo +by(Ettex= Ed Lee eer ‘We can write the result inthe form =c,a(x) +c,B(x)+7(x) where a(x)= Yk*x""**/(mp +2m—1)(mp +2m) Bix)=yk®x"?"78*!/(mp +2m)(mp +2m+1) x)=, (/2)ek"x°"°7=" /(mp+2m+1)(mp +2m=2) We can use theconditions u(1)=u(—1)=0toevaluate¢,andc, c,a(1)+¢,B(1)+ 71)=0 ¢,a(-1)+¢,B(-1)+(-1)=0 sothat «,-BUH=BD) *@(1)B(-1)- B()a(=1) =SD) BONL*@(1)B(-1)- Ba)a(=1) (Thecaseg=2.k=40,p=1wasverified toseven-digit accuracy.) APpUCATIONS inPursics 27 NONLINEAR SCHRODINGER EQUATION: iu,+2ubf +u,, =0 can bewritten iLu+Nu+Lu=0 where L,=9/01, L,=0?/Ax?, andNu=2ujul’. Wecansolve either forthe LoperatorortheL,operator. IfwesolveforL.u,wegetimmediately u=u(x,0)+iL;'Nu+iL;Lu ty=u(x,0) u,=iL7'Ay+iL;'L,uy u,=iLJA, +iLj'L,u, uy, HLA, +iL/Lu, and¢,=0")u,asann-termapproximation converging tou.Ifwesolvefor L,u, wehave u=u,-iL) Lu-L Nu u,=a+Bx 4&4 =I Lu,-LA, forn>0.Thegivenconditions determine o.,.Ifinsteadofafiniteintervalwehavealimitsuchasu->0.asx—>, thenwesetlimu=0 andfindthe limit ofthe series. TIME-DEPENDENT GENERALIZATION OF THE YUKAWA-COUPLED KLEIN-GORDON SCHRODINGER EQUATION: Viv-av/ar-V+|yf =0 VV+idy/dt+Vy=0 Theequations involve operators inx,y, zandt.Ourspecific solution depends onthegiven initial/boundary conditions. Considering theequation forV,and assuming conditions ontare given, 328 Curren 14 L.V=V°V-V+l|yP V,=A+Bt VenSLIVV, -LV,+A,{lt} The AandBmust satisfy given conditions, whether initial orboundary orlimit conditions atinfinity. The Vequation, now using L;=d/dt, results in x=ivy+iv?V Wo=WXY.z,t =0) VouSik VV,+iL)A,{Vy} Tosolve thesystem wefind(y,,V,), then (y;,V,), then (y2,V;), etc. Difficult integrations will arise because ofdifficult initial conditions forwhich solution isdesired, This problem essentially vanishes byapproximation of these functions byafew terms ofanequivalent series, e.g., aMaclaurin se- ries, oFdecomposition ofthefunctions toarrive atelementary integrations. THE N-BODY PROBLEM: Theproblem oftheinteraction ofNbodiesisimportantinmanyconnections andformany force laws which caninvolve attraction orrepulsion, oreven collisions. The problem issoluble bydecomposition because thenonlinearity ccanberepresented bythegeneralized polynomials (Chapter 3)which canbe calculated. Here. wewill consider N-body dynamics inagravitational field Assume asystem ofNpoint masses m,where i=2,3,...N, with positions specified byvectors ¥,fromachosenorigin.Thedistancebetweenm;andmj willbedenoted by#,,=f,-@[=((,-#)-(-7)]"” wherethemultiplication indicates «scalar product. ‘Thetrivialone-body caseisdescribed simplybym,i,=0withinitial conditions F,0)=p, 7O=y Two BODIES. Fortwo bodies inagravitational field, wehave two coupled equations Arricarions nParsics 529 mg”=-os,Fral fel with initial conditions #(0)=3, #0)=p; %O)=7, %(0)=, noting that,2/f,a) =#2/f,.| where, istheunitvector. Wecanfor example, letonemass beasufficiently large, sayM,tobeassumed asafixed origin, then find themotion ofthesmal] mass m.Or,theorigin canbethe observer onearth considering theeffect ofthesunonthemoon. Or,finally, theorigin canbeinarocket traveling through thesolar system, THREE BODIES: For three bodies, wehave: fal Ral -” m,m,_ .M, mM,_mi =-G 1h, -Go a,Fal fel3”gtmM_ogmym_m5oer GSE, Bu Foal with initial conditions: IO=F 2O=7%, BO=7, FO RO=% HO=¥ Four BODIES: ‘Wenow have fourcoupled equations andthecorresponding initial conditions toconsider. 330 Cuarren 14 mf=-GDs, GOs, orat fal fel Fl mf,”=-GETp,-GHaMy GBB,| Fal Fal Fal mf,=-GT Tis,-GDan, Gn,fal ful Bul mi, =-Gs,, Gs, oon,Fas Frasl fish with initial conditions with %(0)=B,and%(0)=,withi=1,2,3,4. GENERALIZING TO N-BODY DYNAMICS: Inagravitational field, wefirstsimplify notation bywriting FG,.5)=@ EG -#)-@ -7)P? which isaconvenient form foruseofthegeneralized A,forf(u,v) discussed inChapter 3.The indicated multiplication isascalar product. Hence forN bodies. ” &mf -7,mz,=-Gm>,aot iF =-Gm,>) m,fG.i) ot LetL=d*/dt? towrite theleftsideasm,L®, andapply L*,atwo-fold definite integration from0totie.,|()état,tobothsides.Alsolet i=xH?andz=fod Ex} Wecanidentify ¥”asthesolution ofLi{" =0,or, FO=P =O)+tFOC =0)=p,+1y, Analogously, = D+, Arpucarions wvparsics 33 The first decomposition component isdetermined and thefollowing components arecalculable from #=-L"GY, m,Ao{f(,7)} #=-LGY. m,A,{F6,5)} Thus n20,for x Bi=-L'GY, m,A,{f6,7)} ff soallcomponents arecalculable bydetermining theA,using themethods of Chapter 3.Forconvenience, thenecessary quantities arelisted The A,foru"™ are: Ay=u," A,=-mu*"u, A,=(1/2)m(m +Dug"? uj=muy?"u, A,=-(1/6)m(m +1)(m+2)u5**? u} +m(m+ Duy? uu,—mu;**? u, TheA,forf(u,v) aregiven by A= DYcluvindf,, where = 332 Charen 14 on Spe=u, ¥, no=Srytoe) Ao=foo Ay=Uifie +Vibe a AsmWah+Val+Gthae 2 Mi PUYfho Ay=Usfig +V5fo+Usfoo +furvs tev] +Vivafos we tt+5phe+a"fy volat Phas Forscalarproblems, f(r)=r7= "A, Agahe A,=7215, A,=3y'f -2G A=45°) -6r- 25 Forf(r)=r? =", A, A= A,=-31‘f, A,=6591-35 A,=-104g6 r?+1215 1,1—31t Forthenonlinearity f(r,,r,) wedefine thenecessary polynomials byA", Aprucarionsmyparsics ee] i@-H)=AY Since“ ¥@-#)=G-a[@-z)-@ -2)]"” =EG,E)nG. i) Wecannowwrite BG.1) =,-1)= a? nG5)=),bY sothat- ~ ~ ~ 4byao={$«|5ot} or) asd a0 J which can berewritten as aeaSweyno Wecandecompose the7,and7,thus: i= le a 5-Le Then= BGA) =G-H= >@-2)=y wv Es a hG.E)=vy?= Since thehfunction isanonlinear function oftwo variables, wecan write itin termsofthegeneralized polynomials A,{f(u,v)}inChapter 3(andalsoinref-erence (4)). Thus . one, ct FQ ae 334 Cnarren 14 ‘Thus wehave asystem ofNcoupled nonlinear second-order equations, or Volterra integral equations when theL"'operator isapplied, which aresolv- able bythedecomposition method since thenonlinearity, though difficult, can beexpressed bymethods ofChapter 3andreference [2].Details forspecific special cases willappear inaforthcoming paper asthey areprogrammed and calculated. NONLINEAR RELATIVISTIC PARTIAL DIFFERENTIAL EQUATIONS: Aclass ofnonlinear equations occuring inmathematical approaches toele- mentary particle theory [5]isgiven by Vo-Fg/dt =mo+s9° (m,g>0) Theobjective istoobtain anon-perturbative time-development ofthefield of this andsimilar equations involving other nonlinear interactions, without the useofcutoff functions ortruncations. Indecomposition form [2-5] (L,=L,+L)o-L=m'9+ eg" where L,=0°/dx*,L, =0°/dy*, L,=07/027, L,=-07/at". Thepossi- bleoperatorequations forpartialsolutions {4]are: Ly=m*9+e9°-(L,+L,-Lo L,o=m'g+g9'-(L, +L,-Lg L.o=m'o+g9'-(L,+L,-L)o L.y=-m’g-gg°+(L, +L,+L,)o Thesolution ¢({x.y,z.t) willbeapproximated bythen-term decomposition series ®, 0,=Fey, Wevisualize amanifold inafour-dimensional coordinate system with x,y.z,t axes. Onthecoordinate planes wehave curves 9(x), p(y)@\2).and @(t) rep- resenting intersection softhe@“surface” with theplanes. Wecanstart from any ofthese functions (i.e., theinitial-boundary conditions) togenerate y(x,y,2,1). Wecanbegin with anyoneofthese equations toyield thesolution Aprucarions wPatsics 335 and ingeneral alloftheresults areidentical. (Inspecial cases, they are asymptotically equal.) Suppose weconsider thefourth equation. Then operating withL;'orthe two-fold definite integration from0totandidentifying theinitialterm =O =0, X,y,2)+t9"(t=0, XY.2) wehave 9=Q+Li mg-L}' gg’+L)'[L, +L,+L,19 Wenow apply thedecomposition 9-L% = Wecannow determine allcomponents of@;thus, 9=-L' mg, —L;'gy+Li'(L, +L,+L.) 9,=-Lim’g, -LigA,+LL, +L,+L)9, an=—Liim'p, -Li'gA,+LUL, +L,+1.)Pq Now = J isaconvergent approximation to@=)”, @,,i.e.,tothesolution. Which of thefour equations weusedepends onwhich initial/boundary conditions are bestknown ormeasured, When thenonlinearity isofhigher degree, e.g.inthe scalar relativistic equation: Vip-(F dt )p=m’p+ gy? thesameprocedure applieswiththeappropriate A,forg?forwhichrulesare given in(2) TIME-DEPENDENT SCHRODINGER EQUATION INCONFIGURATION SPACE: The equation is(6] 336 Cuarren14 2-Lvvenev@ven=2% 2m iat ThevectorFisthepositionvectoroftheparticlereferredtoaconvenient ori- gin.Wecanintroduce unitvectors along axes ofrectangular (x,y,z) orspheri- cal(r,8,0) coordinates. Interms ofrectangular Cartesian coordinates, theop- erator Vi=L,+L,+L, where L,=d*/d?, L,=day", L,=Fa’. Then 2mav2m L,+L,+L,jy=2 Sy [Ltt +h.)iha Using decomposition, wecansolve foranyoftheoperators L,,L,,L,, or L,=d/dt aslong asweknow theappropriate conditions onx,y,2,ortThe inverse operators L;',Ly’,L;'aretwo-fold indefinite integrations respectively inx,y,2,The inverse L;'isasingle definite integration from zero tot representing aninitial condition problem forwhich \¥(t=0)isrequired, Inthe other cases. wehave boundary-value problems forwhich weneed values of Wattwo values ofxoryor2.SupposewesolveintermsofL,,then L,W=0%«pve-L,Y-L.Y where &=2m/ih andB=2m/h?. Then wemust operate onallterms with Lj).The leftsidebecomes L;'L,¥ =¥- A-Bx. Rearranging, wehave Y=A+Bx-aLy a/at¥~ BL]VY-L;[L, +L,J¥. LeeY=DO,Y, andidentify,asA+Bx.Thenann-termapproximant toYdenotedby@, will be ol¥l= 2DYe which becomes ¥inthelimit as nee. Because ofrapid convergence, a few terms aregenerally sufficient. (When thisisnotthecase, onecanusePadé approximants orother acceleration techniques orthemethod ofasymptotic decomposition.) The decomposition components are Armucarions wyPursics a7 Y,=A+Bx ad = =Healy oy-BLVY,-LIL,+L.)% w=au, -BLVY,-Li[L,+L.) ¥,soL24, BLVY,-Li[L, +L,}¥. andwecannow write ©,[‘]. Foraparticular potential, e.g.anisotropic harmonic oscillator, V=1/2mw*(x’ +y?+z"), wecannow calculate the solution, Evidently, wecanalso deal with nonlinear potential functions or nonlinear Schrédinger equations, The decomposition method isnotrestricted topotentials varying slowly inadeBroglie wavelength. The problem solution iscomplete when AandBareevaluated bymatching tothegiven conditions. This requires matching the@,totheboundary conditions foreach value ofnaspreviouslydiscussed, Other interesting examples (Ginzburg-Landau equation, Euler equations for inviscid flow, isentropic flow) aswell asfurther results onNavier-Stokes ‘equations andonnumerical computation willappear infuture publications, REFERENCES 1. G.Adomian, Analytic Solution oftheStochastic Navier-Stokes System, Found. of Physics,21,(831-843)(July1991). 2. G.Adomian, Nonlinear Stochastic Systems Theory and Applications toPhysics, Kluwer (1989). 3. G.Adomian, Linear Random Operator EquationsinMathematical PhysicsI,II,1, J.Math. Phys., 11,3,(1069 -1084) (1970), 12,9,(1971), (1944 -1955). 4. N.T. Shawagfeb, Lane-Emden Equation, J.Math. Phys., 34,9,(4364 -4369) (Sept. 1993). 5. I.Segal, Quantization andDispersion forNonlinear Relativistic Equations, inThe Mathematical TheoryofElementary Particles, R.GoodmanandI.Segal(eds.),MIT Press (1965). 6. DS. Saxon, Elementary Quantum Mechanics, Holden-Day (1968). SUGGESTED READING 1A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics I-II, J.Lumley (ed.), MIT Press (1971). 2. LD. Landau and EM. Lifsbite, Quantum Mechanics, J.B.Sykes and J,8.Bell (transl.), Addison-Wesley (1958). 3.LD. Landau and E.M, Lifsbite, Mechanics, J,B.Sykes andJ.S.Bell(trans.), Addison-Wesley (1960). APPENDIX I PADE AND SHANKS TRANSFORMS PADE APPROXIMANTS: Theobjective hereistofindasolution inthelarge, i.intherange (0,<) from thedecomposition series which normally hasafinite circle ofconver- gence forinitial-value problems. Theprocedure istoseek arational function forthe series. Given afunction f(z) expanded inaMaclaurin series f(z)=O~, ¢,2". wecanusethecoefficients oftheseriestorepresent the function byaratio oftwopolynomials a,taz+--taz" b,+b.z+--+b,2" symbolized by[L/M] andcalled thePadé approximant. The basic idea isto match theseries coefficients asfaraspossible. Even though theseries hasa finite region ofconvergence, wecanobtain thelimit ofthefunction asx—© ifL=M. Notice thatifwearesatisfiedwith[1/1].wewillhave (a+a,2)=(b, +b2Nlc, +024E24...) sothatcoefficients of2°arezero, ie, bic, +bye; =0 Taking by=1,wehave bc, + =0 Now consider [2/2] or (ay+a2+8:27) =(Dy+biz+baz"Cy+C2+C2?+C2+--+) Clearly thecoefficients of2’arezero, sothatwecanwrite bac; +bye; +byes =0 Ingeneral, wenote that there are L+1independent coefficients inthe numerator and M+1coefficients inthedenominator, Tomake thesystem determinable, itiscustomary tolet_by =1,We then have Mindependent 338 PaveavoSHANKSTRANSFORMS 339 coefficients inthedenominator andL+M+|independent coefficients inall Now the [L/M] approximant can fitthepower series through orders 1,z,2?,...,24™ with anerror of0(z*"), Forexample, for Lila f(z)=1-s24a2? +(2)=1-p2432 we have 1+(/6)z 2 Wj=———= =f(z)+of N=Tape712)*0’) Consequently, (ao+ajZ+ +az") =(Dp+D,Z + +DYZ™(G +E,Z+---) Equating coefficients ofz*',z**,..., 24 intum, wecanwrite Duicrater +DserCusiga too+BoC =0 DaCosta +DaeiCeases +o>+Daca =0 bye+Dreier tos+DocLyy=0 Setting by=1,wehave Mlinear equations forthe Mcoefficients inthedenom- inator. Cuemet Stamos CL Dy fern CumSemeesCLs||Paer||Cuee c oy Cusaatd LOy Cue, ‘Weinvert thematrix ontheleftandsolve forthe b,fori=1,..., M.Since weknow theCo,Ci,Cay.» WECanequate coefficients of1,2,2%...,2to BOt Ay,5,..45 a4. Thus y=Cp a,=e,+bic a,=¢, +b, +b,c aint. aaaqt Yd 340 Avpenoee1 Thus thenumerator anddenominator ofthePadé approximant aredetermined andwehave agreement with theoriginal series through order z“*™, From the matrix equation, wecanwrite thelower-order approximants. (For higher or- ders, one canusesymbolic programs.) ) fMj= 01) 2) MMJ=(22] b= -G %6)(b:)__(% &&)\b) le or »,)_(G/D -e/P) (-c,db)"-qDa/DJ\-c whereD=¢,¢,—c}sothatwehave csc,-¢? b,=SST Fee b,=Sf78ies ee Forf(z)=cy+cz+¢,27+--.wehave Gn)eBoth (ijj=-2— lim{i/1] =a,/b, l=ee ;Jim[ya]=a/ [oya}=Bete ase lim[2/2}=a,/b, PPIoaths Jim[2/2}=a,/b, a,+az+a,2°+a,2° gp3}= eterae+2 Jim|3/3]= a/b, [3/3]Secrets tim[3/3]=a,/b, tim[m/m]= a,/bx EXAMPLE: Find thelimit fore°“('**) (1) =.333... [2/2] =368... [3/3] =.368... EXAMPLE: For e*,we have Pave ano Suants TRANSFORMS aa 2ex y=22%W}=s= 1246x42 2/2)=$ =Carer ees 120+60x+12x"+x" 33)=$ee (l=90 ¢0x+ 1a = Note thatifweletx= toconsider theseries fore,wegetthecorrect limit, (wn =3 (22) =2.714 (3/3) =2.718 [4/4] =2.718 PROBLEM: Noting that thelimit iscorrect forx=1butthelimit at©for (1/1), (22), {3/3} fluctuates between +Iforboth e*and e%,i.e., ag/b_=+ 1 asmincreases, explain thelack ofconvergence toalimit since weknow e*0asxo0ande*$00asxo,TrytheShanks(orWynn") transformation. These transformations areeffective inaccelerating convergence ofmany slowly convergent series. Forcases where thePadé approximant appears inapplicable, wecansome- times use transformations oftheseries. Consider H(2)=Cy+yz+Caz+oo 1.04,Cu#0DUECres=Cra2=0.Welety=z?sothat fly)=cotoy+ery?+~~ isnow inaform forPadé transformation. Ingeneral foraseries f(@)= Dcya” where Nisapositive integer (or“skip factor”) with cy,#0butCyq., =0 for1sv<N-1.Wesubstitute y=Z"toget *Other useful transforms used toaccelerate convergence aretheEuler, Wynn, andVan ‘Wijngaarden transforms 302 Arrenoa1 f(y)=Co#Cy+Cony?++ or fy)=d ay" where n= Nm,- 2exampce: —f(x)=(22@2%)" 21-3542 2H...T+2x 4°" 32 Toapproximate by[/1] wehave cbr =—¢: 3,38 4732 b= 138 p=Cy=13B =¢,+b¢)=-2+ B12aseebe=-s+E 1s78 Consequently1+ _ 1)=2UB_054 Wlgap (which iswithin 8%ofthecorrect limit of0.5forthefunction). VERIFICATION: 6B, 3,392) 102 5Lex} 1-Sx-dex? =1+2x=Ox gage ynttgrr) EXERCISE: Calculate {2/2}and(3/3}toshowthatthelimitapproaches 0.5 more and more closely aswegotohigher-order [L/M]. EXAMPLE: f(x) =1—x+x7/2! -xB!+ 1-(/2)synj=tO2s |, « Ww)Tax 8*> EXAMPLE: f(x) =e* Pavé ano Snans TRANSFORMS 383 2+xy=228== 1246x+x° Pl12-6x+x° EXAMPLE: fle2x7* 1.5.,,13., M1,10"Tex|mle g*tig8" We note that even though theseries hasalimited region ofconvergence (x<12),thefunctionissmoothfor0<x<o,Ifwewritearatio (a+bx) /(c+dx) itisclear that wegetafinite limit asxapproaches 2, Calculating (L/M] = (U1, wehave 1+(7/4)xY= 14 ~(vjGaye Mee andcarrying outthedivision, weget 1 5.,25 5125 4qjeisdx—-3e Bye Bey...fy] tpxcge type age+ whichexactlymatchesthefirstthreetermsoftheTaylorseries.Ifwegoas high as[5/5], wematch thefirst 11termsoftheTaylorseries 1+(13/4)x+(41/16)x? 2/2)=Oe+CTO +1.4137 21)= aiiax+ asx>MB7x which isthecorrect limit of2'tothree-place accuracy. EXERCISE: Sincecosx=~,(-1)*x/(2n)!, showthat [2/2]=(12-5x*)/(12+x?) uses thefirst five terms oftheTaylor series, Show that [2/2] iscloser tothe exactvalueofcosxthanthesumofthefirst5terms. ata ‘Aprexon| SERIES NOT SUITABLE FOR PADE TRANSFORM: Sometimes the series isnot inaconvenient form for the Padé transform which isdesigned foraseriesJ)",cx”withnon-zero CyCy,C2. Ifwehavemissing terms, ¢.g.,(x)=D.™,Cys where Nisapositive integer andcy#0,€.24 DingCOX?+Gx?+c,x°++thenweusethetrans- form z=x"(orz=x° inthisspecific case) andletb,=c,,towrite £(z)=D7,b,2*andb,#0which isnowsuitable forthePadétransform. If£(x)=D7,ox"andcy=0,wecanapplyatranslation z=x-Ewith &<p with psymbolizing theradius ofconvergence. Ifweuse|{|<1,the result will besimpler formanual computation because theresulting series for cachnewcoefficient inf(z)=.™,b,z*willconverge rapidly. Theresultfor b,willbeb,#0and alvt" b= C06yon) which isequivalent toananalytic continuation. THE SHANKS TRANSFORM: This isanonlinear transform which can bevery effective, particularly in accelerating convergence ofslowly converging series. Ithaseven been applied todiverging series which seems contradictory. However, ifapowerserieshas been obiained bydividing outarational function, thisnonlinear transform isa meansofinvertingtheprocedure10obtaintherationalfunction. ‘The Shanks wansform isrelated tothePadé approximant. Itismore accurate: however, thePadé approximant ismore explicitly expressed interms ofthe coefficients oftheoriginal series. Lets write thesequence ofpartial sums {S,) fora series anddefine theShanks transform by T{s,}=SSeS©Spar Sy>2S Weoften want repeated transforms, called theiterated Shanks transform, soit isconvenient towrite {A} forthe{S,}. The iterated transforms will ofien lead (oanextremely accurate solution. Thefirst-order transform iswritten as: PADE AND SHANKS TRANSFORMS a5 B,=T(A,}=—AssAsr AsAy tAl 72A, C.=T{B,} D,=T{C,} ‘Wenow write thesequence simply as Ag A,By ArBr Cr AsB,C, Ds where A,=S,. EXAMPLE :Inthecontinued fraction representing YZ,thesequence of partial sums {S,} isS,=1, S,=3/2, S;=7/5. Find thelimit. Calculation yields: eeeea [2[32 {ime [tg99r70_|19601/13860, [a|)|s7708[| Ls [ang | We note that As=1.4137 which iscorrect to3figures, while C,= 1.414213564 iscomect to9figures. EXERCISE: Verify allresults. EXAMPLE: The Leibnitz series for mism= 4-4/3 +4/5 -4/7 +.... The results are 346 Arrexon.1 [0[s.cooo00] fT[1 [2.666667 [3.166667] | 3.466667|31333333|3.1421053[|S [3|2.8950381 |3.1452381 |3.1414502[3.1415993| |[4_|3.2396825 |3.1396825 |3.1416433 |3.1415909 |3.1415928 2.9760462|3.1427129|3.1415713|3.1415933|3.1415927 [6|3.2837385 [3.1408814 [31416029 |3.141s925[ i 30170718 [31420718 [3.i4iss73[ | [s_[3.2823659[ssaiasaa[ TT [o_[s.osisz06 | Weseethat thetenth partial sum Ag,orSe,iscorrect only toone figure. Shanks [1]points outthattogive ananswer correct toeight figures would require n=40million inS,while wenote that e,(S,). orE,,isalready correct toeight figures. EXERCISE: Intheexample 3.39: f(xje13x +22x8- Wal gx3 intheprevious section onPadé transforms, weobtained [1/1] =.54which was close tothe correct limit of.S.ShowthattheShankstransformT(S;)= .54also. Investigate therelationship between thetwoprocedures. Another related transform (which can also beiterated like the Shanks transform) istheAitken transform defined by T{S,}=S, -{(S,.,-S,)°MS,-28,.,+S,)} Thefirst-order Shanks transform isequivalent totheAitken (67)process and themth order Shanks transform ofthenthpartial sum isequivalent tothe[rm/n]} Padé approximant. Wecanview theShanks transform asaunifying concept subsuming theAitken process andthePadé approximent. PADE Avo SHANKS TRANSFORMS 347 SUGGESTED READING 1, Daniel Shanks, Nonlinear Transformations ofDivergent andSlowly Converging Series J.Math. and Physics, 34,(1-42) (1953). 2. A.C. Aitken, OnBernoulli's Numerical Solution ofAlgebraic Equations. Proc. Roy Soc., Edinburg, (289-305) (1926). APPENDIX IT ON STAGGERED SUMMATION OFDOUBLE DECOMPOSITION SERIES uy=ul” a,=u + a=u) +n! +a uy=u?) +ulul +ul? uy=ul+ul)+ul)eu? +0? Thefirstcolumn ontheright oftheequation isequal tous.Thesecond column isequal tou;,The third column isequal tou;,etc.These sums ofeach column willbedenoted byu}, uj, u}.... respectively where iindicates theinitial- value format. Thesumofthecolumn tothelefioftheequal sign isdenoted by SE,uhforboundary-value format. Wehave > u=D) ubsujtul+.= Yul ‘Thus inwriting theapproximants 0,wecanwrite o[u*]=¢,[us] 4,lu]=o,{us]+0,[u)] afu’}=o,[u,}+e,fui]+9[u;] al")=4-[us]+4.[ui] +--+[42-1] which can bewritten Ge{0"]= XG0-a[4s] (2) Referring again to(1),wehave Sragcenen Suuarion 349 ag=Ue)+ul? eu pee uD,+ul,eu? oru,=", ue"inourboundary-value format. Retuming to(1)wecan write ininitial-value format, uu,=u) +uf+uf+ =u +uluf 4 uy=u +ul+uf)4 oru,=5,uf?(ininitial-value format). Bydecomposition u=™,u,. a a -v” te), Bydouble decomposition u=>", Y-, ul”. FORMULAS OF INTEREST: ELw-E Ew a mm ea[u"]= Y9.-.[4.] Fd a> yo wey ufPs Let youl=>uf wad Sue” cand ms 2% w=u' Jim,[u*]=u° limY4.-.[us]= >uh=u"mon ano and , APPENDIX III CAUCHY PRODUCTS OF INFINITE SERIES Inonedimension for=>, a,x’andB=zr,Bjx®,wecanwrite Bu->«xBana Intwo dimensions, Inthree dimensions,; Infour dimensions, sso aver Prooucrs oF Inewire Senet 331 bus YY YY xexpghx =O ge ge gadm= InNdimensions, wed Say tox These canallbeprogrammed using “do-loops” and stopping rules. These product rules should beuseful inprogramming solutions where thesystem input andsystem coefficients areknown only aspower series. INDEX Mixed Derivatives 46 Acceleration Techniques 30,206, 338 Modified Decomposition 115, 131, 138, (Adomian) Polynomials Reference List 214 1 ‘Neumann Boundary Conditions 190 Analytic Simulants 17,289 Noise Terms 29Applications NonlinearPartialDifferential Equations‘Advection 316 80 ‘Advection-Diffusion 318 Nonlinear Ordinary Differential EquationsBurger'sEquation311 85Dissipative Wave Equation 28,30 Nonlinear Oscillations 228, KAV Equation 321 Partial Solutions 23,28.30,32,35.36Kuramoto-Sivasbinsky Equation312 Perturbation 284Lane-Emden Equation 315 Proliferation ofTerms 254 N-body Problem 328 ‘Smooth Expansions ofPiecewise- Navier-Stokes Equation 302 Differentiable Functions 298 Nonlinear Heat Equation 322 Spatial andTemporal Formats 97, Nonlinear Kiein-Gordon Equation 322 106.109, Nonlinear Relativistic Partial Staggered Summation 243, 348 Differential Equation 334 Stopping Rules 18 Nonlinear Transpor inMoving Fluids 316 Random Nonlinear Heat Equation 323 ‘Schrddinger Equation Nonlinea: 327 Quartic Potential 325 ‘Yukawa-coupled Kiein-Gordon 327 Sine-Gordon Equation 324 ‘Turbulence 367 Van derPo!Equation 230. 231.308 Applications ofModified Decomposition 154 Asymptotic Decomposition 18.241 Boundary Conditions atinfinity 211 Boundary-value Problems 87, 114. 138, 288 Cauchy Products 350 Convergence Regions 23.25 Decomposition forordinary differential equations 6.28 forpartial differential equations 22,28 Difficult Nonlinearities 150 Dirichlet Conditions 75 Double Decomposition 22.69,87 Duffing Equation, 154, 157, 230, 231, 235, 236. 263, 277, 280 Generalized (Adomian) Polynoesials SO Generalized Taylor Series 10 Gibbs Phenomena 301 Harmonic Oscillator 247. 251 Integral Boundary Conditions 196 Integral Equations 224 lnegular Contours orSurfaces 288 Fundamental Theories ofPhysics 22. AO. Barut and A.vanderMerwe (eds,): Selected Scientific Papers ofAlfred Landé. [1888-1975], 1988 ISBN 90-277-2594.223.W.T.Grandy,Jr:Foundations ofStatistical Mechanics. Vol. I:Nonequiliorium Phenomena. 1988 ISBN 90-277-2649-324,ELLBitsakisandC.A.Nicolaides (eds.):TheConceptofProbability. Proceedings ofthe Delphi Conference (Delphi, Greece, 1987). 1989 ISBN 90-277-2679-5,25.A.vanderMerwe,FSelleriandG.Tarozzi(eds.): Microphysical RealityandQuantumFormalism, Vol.1.Proceedings oftheInternational Conference (Urbino,Italy,1985).1988 ISBN90-277-2683-3, 26.A.vanderMerwe,FSelleriandG.Tarozzi(eds.):Microphysical RealityandQuantumFormalism, Vol. 2.Proceedings oftheIntemational Conference (Urbino, Italy, 1985), 1988 ISBN 90-277-2684-1 27, LD. Novikov and V.P.Frolov: Physics ofBlack Holes. 1989 ISBN 90-277-2685-X 28. G.Tarozzi and A.vander Merwe (eds): The Nature ofQuantum Paradoxes. Italian Studies intheFoundations andPhilosophy ofModern Physics. 1988 ISBN 90-277-2703-1 29. BR. Iyer, N.Mukunda and CV. Vishveshwara (eds.): Gravitation, Gauge Theories andtheEarly Universe. 1989 ISBN 90-277-2710-4 30, H.Mark and L.Wood (eds.): Energy inPhysics, War and Peace. AFestschrift celebrating Edward Teller’s 80th Birthday. 1988 ISBN 90-277-2775-9 31, GJ. Erickson and C.R. Smith (eds.): Maximum-Entropy and Bayesian Methods in Science and Engineering. Vol.I:Foundations. 1988 ISBN90-277-2793-7 32. GJ. Erickson and CR, Smith (eds.): Marimum-Eniropy and Bayesian Methods in Science and Engineering. Vol. It:Applications. 1988 ISBN 90-277-2794-533.MEE.NozandY.S.Kim(eds.):SpecialRelativityandQuantumTheory.ACollection ofPapers onthePoincaré Group. 1988 ISBN 90-277-2799-6 34, L¥u, Kobzarev and Yu. Manin: Elementary Particles. Mathematics, Physics and Philosophy. 1989 ISBN 0-7923-0098-X 35.F.Seller:QuantumParadoxes andPhysicalReality.1990_ISBN0-7923-0253-236.J.Skilling (ed.): Maximum-Entropy and Bayesian Methods. Proceedings oftheSth International Workshop (Cambridge, UK, 1988). 1989 ISBN 0-7923-0224-937.M,Kafatos(ed.):Bell'sTheorem,QuantumTheoryandConceptions oftheUniverse.1989 ISBN 0-7923-0496-9 38. Yu.A, Izyumov andV.N. Syromyatnikov: Phase Transitions and Crystal Symmetry. 1990 ISBN0-7923-0542-6 39.PLP.Fougtre (e4,): Maximum-Entropy andBayesian Methods. Proceedings ofthe9th Intemational Workshop (Dartmouth, Massachusetts, USA, 1988). 1990 ISBN 0-7923-0928-6 40.L.deBroglie: Heisenberg's Uncertainties andtheProbabilistic Interpretation ofWaveMechanics. WithCriticalNotesofthe Author. 1990 ISBN 0-7923-0929-4 41,WT.Grandy,Jr:Relativistic QuantumMechanics ofLeptonsandFields.1991ISBN 0-7923-1049-7 42.YuL,Kiimontovich: TurbulentMotionandtheStructureofChaos.ANewApproachtotheStatistical TheoryofOpenSystems.1991 ISBN0-7923-1114-0 Fundamental Theories ofPhysics 43.W.T. Grandy, Jr.andLH. Schick (eds.): Maximum-Entropy andBayesian Methods Proceedings ofthe10th Intemational Workshop (Laramie, Wyoming, USA, 1990) 1991 ISBN 0-7923-1140-X 44, PPtdk and S.Pulmannové: Orthomodular Structures asQuantum Logics. Intrinsic Properties, State Space andProbabilistic Topies. 1991 ISBN 0-7923-1207-4 45. D.Hestenes and A.Weingartshofer(eds,):TheElectron.NewTheoryandExperiment. 1991ISBN 0-7923-1356-9 46. P.PIM. Schram: Kinetic TheoryofGasesandPlasmas.1991ISBN0-7923-1392-5, 47, A.Micali,R.BoudetandJ.Helmstetter(eds.):CliffordAlgebrasandtheirApplications inMathematical Physics. 1992 ISBN 0-7923-1623-1 48, E,Prugovedki: Quantun Geometry. AFramework forQuantum General Relativity1992 ISBN0-7923-1640-1 49. MH. Mac Gregor: TheEnigmatic Electron, 1992 ISBN 0-7923-1982-6 50, C.R. Smith, GJ. EricksonandP.O,Neudorfer(eds.):MaximumEntropyandBayesian Methods. Proceedings ofthe 1th International Workshop (Seattle, 1991). 1993 ISBN 0-7923-2031-X SI. DJ. Hoekzema: TheQuantum Labyrinth. 1993 ISBN 0-7923-2066-252.ZOriewicz,B.JancewicaandA.Borowiec(eds.):Spinors,Twistors,CliffordAlgebrasand Quantwn Deformations. Proceedings ofthe Second Max Bom Symposium (Wroclaw, Poland. 1992), 1993 ISBN 0-7923-2251-7 53. A.Mohammad-Djafari and G.Demoment (eds.): Maximum Entropy and Bavestan ‘Method Proceedings ofthe12th Intemational Workshop (Paris. France, 1992). 1993, ISBN 0-7923-2280-0 54. M.Riesz: Cliford Numbers and Spinors with Riesz” Private Lectures toE.Folke Bolinder and aHistorical Review byPerti Lounesto. E.F. Bolinder and P.Lounesto (eds. 1993 ISBN 0-7923-2299-155.F,Bracks,R,DelangheandH.Serras(eds):CliffordAlgebrasandtheirApplicationsinMathematical Physics.Proceedings oftheThirdConference (Deinze,1993)1993ISBN 0-7923-2347-5 56.1.R.Fanchi:Parametrized Relativistic QuantumTheory.1993ISBN0-7923-2376-957. A.Peres: Quantum Theors: ConceptsandMethods.1993 ISBN0-7923.2549-4 58. PL. Antonelli, R.S. Ingarden and M.Matsumoto: The Theory ofSprays and Finsler Spaces with Applications inPhysics andBiologs. 1993 ISBN 0-7923-2577-X 59. R,Miron andM.Anastasiei:TheGeometryofLagrangeSpaces:TheoryandApplica tions. 1994 ISBN 0-7923-2591-5 60.G.Adomian:SolvingFrontierProblemsofPhysics:TheDecomposition Method.1994ISBN 0-7923-2644-X KLUWER ACADEMIC PUBLISHERS -DORDRECHT /BOSTON/LONDON