dini schwartz

IndianJ.pureappl.Math.,19(11):1089-1100November, 1988 THE MODIFIED DINI’S SERIES AND THE FINITE HANKEL-SCHWARTZ INTEGRAL TRANSFORMATION J.M. Menvez Departmento deEcuaciones Funcionales, Facultad deMatematicas, Universidad deLaLaguna, LaLaguna (Canary Islands—Spain) (Received 5August 1986; after revision 25April 1988) Inthispaper, anarbitrary function f(x) defined onthe interval (0,a)is ‘expressed as an expansion inDini's series of the orthogonal family (Fs (om2)ofmodified Bessel functions, where Gy (x)=x-¥Ji(x)and pm denotes the mth postive root oftheequation x$F",(ax)+hF,(ax)=0. Next, the convergence theorem isrigorously established. The Din's seriessuggesttoconsideravaraintofthefiniteHankeltransformation, whichwilbe called the finite Hankel-Schwartz transformation ofthe second kind. This transformation isusedinsolvingsomepartialdifferential equationswhich‘cannotbedirectly treated byapplying thecorresponding finiteHankel trans- formation, Fisally, weremark thatthe initial term ofour expansion depends ‘only ontheparameter h,whereas theclassical Dini’s expansion depends on hee 1, Inrropucrion Schwartz’ investigated thefollowing modified Hankel transformation FQ)=FFOO) ams) (lt) where dm(x)=[2°TP(+WpxtdxandF(x)=2°T(+1)x-¥Jy(x),Jv) being theBessel function ofthefirstkind oforder v. This transformation hasbeen studied inspaces ofdistributions byseveral authors and Lee’ calls itHankel-Schwartz integral transformation. Toconsider theexpansion ofanarbitrary function f(x) defined intheinterval (0,a)asaFourier-Bessel series, i.e.,asaseries ofthetype LQ)=Za Fon) (1.2) where Fv(x)=x-’Jy(x),¥ 2—}andJ,denote thepositive zeros ofthefunctions Fi), ie, Fr(ina) =0. (1.3) 1090 J.M.MENDEZ Méndez* introduced thecorresponding finite transformation through theequation Ar(FD) =Faw) =dXEFy(jax)f(x)dx (1.4) which iscalled Hankel-Schwartz transformation ofthe first kind oforder v, Its inversion theorem isstated as: Theorem \—Letf(t)beafunction defined in(0,1)andassumed tobeabsolutely summable over the same interval. Letv =>—4and 7a= er Fy(i,t)f(0)dt,n=1,2,.. KFA.)8 Iff(1)isofbounded variation in(a,6),0 <a<6 <1, and ifx€(a,6),then the series (1.2) converges to LLG + +FO -OL Inthispaperweshowhowthemodified Diniseriesexpansion ofanarbitrary function f(x) leads naturally tothefinite Hankel-Schwartz integral transformation of thesecond kind. The inverson theorem ofthis new transformation isrigorously established bystudying the convergence oftheseries expansion. The operational calculus generated isused inthesolution ofseveral problems inMathematical Physics. Recall that theform oftheDini series isdetermined bythe nature ofthezeros oftheequation 2Fi @M+hF (=O. Weemphasize thatthefirsttermoftheexpansions dependontheparameter /,and ‘notonA+y,asithappens intheclassical theory (cf.Watson", p.597). Another interesting feature ofthis transformation isitsusefulness inthesolution ofpartial differential equations which cannot betackled byapplying thecorresponding finiteHankel transformation whenv#|(cf.Colombo‘, p.82). Finally, wewilldenote inthesequel thepartial sum oftheseries (1.2) by Si(3)~BamFoUm3). 0.5) Setting Pew= Sp2HUes) Fed A)FA) HANKEL-SCHWARTZ INTGRAL TRANSFORMATION 1091 we have Se) =fo PLO,OF(0dt. 0.1) 2. Preciminary Resutts LetLdenotethedifferential operatorx71<2.trvZéWebeginbyconsi- dering thefollowing Sturm-Liouville problem (cf.Sneddon”, p.440) (L+™)y =0,0<a<b (21), My(a)=a,y(a)+ay (a)=0,Ny(B)=b,y(6)+5,9"(1)=0 wo(2.2) where a,,a,,6,and 6,represent prescribed constants. ‘The general solution ofeqn. (2.1) is Y= $9) =40) FAs +BO) RAW (23) where Fy(x)=x-*Jo(x)andYo(x)=x-¥Yo(0),Jo(x)being theBessel function ofthefirst kind oforder vand Y,(x)theone ofsecond kind. Lety=¢,(x)betheeigenfunctions oftheproblem (2.1)—(2.2) which corres- pond tothe nonzero eigenvalues A,. Wehave thegeneral orthogonality condition ifmln"{xODEx93) ’ wo S374,0)bn@)dx=4 +2600) 6i].ifmen t0 ifmzn, (2.4) Then, wededuce from (2.3) that thesolution oftheparticular problem (L+9)¢@)=0, O<xca wns(2.5) N¢(@)=¢'@)+hg(@)=0,4>0 is $n(x)=Fo(P,x) (2.6) whereP,,P,,..denote thepositivezerosarranged inascending orderofmagnitude of thetranscendental equation 2F, (a2)+h F(a) =0 me) ive,fof.Gray etal.',p.16,eqn. (24)), aitFy,(az)+hFs(az)=0. 1092 J,MeMENDEZ The above orthogonality condition (2.4) now becomes t Spe(ah+ab20h) 2(Ppa)ifman[2FP) Fe) dx=}77F‘ Lo vif zn, 28) Given anarbitrary functionf(x)definedintheinterval(0,a),(2.8)allowsoneto formally express thisfunction asaDini expansion, asfollows LG)=Eby Fs(Pmx) (2.9) where 2e% ‘ox= jXESF,(Px)f(x)dx att)FF(Pma)(ah?+apt,—2vh)@ -..(2.10) m=1,2,...,Pmbeing thepositive roots ofeqn. (2.7). Note that wecanextend theDixon theorem (cf.Watson’, p.480) tothezeros ofthefunction (2.7). Indeed, itcan beproved that the zeros ofthe equations AxF(x) +BH (x)=Oand CxF', (x)+DFvix) =0areinterlaced, what- ever thereal numbers 4,B,Cand D,provided they aresuch that AD #BC. Hence, theroots ofeqns. (1.3) and (2.7) also areinterlaced. 3. THe Mopiriep Dint ExPansion—THE CONVERGENCE THEOREM: Inthis section itwill beassumed that a=|forthesake ofsimplicity. Asitoccurs intheclassical theory ofDini expansions (cf.Watson’?, p.597), we have toaddaninitial term totheseries (2.9), Infact, the form ofDini expansion is based onthezeros ofthefunction (2.7) and these depend upon thevalues ofthepara- meter h.Thus, theexpansion (2.9) only corresponds tothecase h>0. When h=0itcaneasily beseen that theequation (2.7) hasazero attheorigin. Ontheother hand, when /<0thisfunction hastwo purely imaginary zeros. Letvbeareal number such that v>—}.Wewrite themodified Dini expan- sion off(x),asfollows £0)=botEbmnFs(Pu) 2.1) where bydenote theinitial term which must beinserted in(2.9) asa consequence of the existence ofthese new roots. HANKEL-SCHWARTZ INTEGRAL TRANSPORMATION 1093 Ifh >Othe initial term 6,=0and (3.1) coincides with (2.9). But when h=0,taking into account that Jx8 (Om3)de=Figs_)=0,m=1,2. [cf.Gray etal.', p.16,eqn. (25)] and (2.7), weget by=r+ 2)|xS00) ds G2) Finally, if+Pyidenote the imaginary zeros of(2.7) when h<0,from (3.1) and (2.8) weinfer that 2°53 2 b=——__—__—_ Jx1 (iPex)f(a)de....8.3)(P+ 2vh—ht)FF (Pei) & Now, consider thefunction 2wFe(aw)Fs(ow) 34) Fw) WH, (w)+h Fy(w)) whose poles arethezeros ji,js... ofFv(2)and thezeros Pi,Ay... of:F, (2) +h F(2). The residues ofthis function atthe first poles are 2FiUe) Flin) BF, Um) Ifh>0theresiduesatthepolesPs,Pay...are 20%, Fv(Pax) Fe(Pa) (PL—2h+ht)F}(Pe) When h=0wemust moreover consider theresidue atthe origin, whose value is 4@ +1). When h<0theresidues at-:ip,areboth equal to 20 Fs(Poxi)Fu(Poti) (P2+2vh —ht)F3(Poi) 1094 J.M.MENDEZ Bydenoting thepartial sumoftheseries (3.1) 44)=be+EbnFo(Pe3) G5) and oa27,Fi(PeaL(Pmt) Po 158) =Ag(a0) + 8.6) fat(P3,— 2vh+At) 2(P,) where {O,ifh>0 |26+ Difh=0 Aen =|" (3.7) DSS apt (Poxi)Fe(Pott) | ith<o {(P+2h—HF)F2QPpi) wocan express (3.5) a8 a,(x)=f09Py(x,A)FU)at. (8) Now choose D,such that itisnot equal toanyofthenumberjaandP,<D, <<Pas, and letjwbethegreatest ofthenumbers j,,which does not exceed D,(cf. Watson", p.598), The following expression 5,6058)=SSPFU Fea) _45x,1) wt inFey (im) ean2PL,Fs(Poex)Fe(0,1)->ans 39)fear(P5,+ ht—2vh)F?(Pn) where A,(x,t)isgiven by(3.7), permits toconnect thepartial sums ofthe modified series ofFourier-Bessel (1.2)and Dini (3.1). Clearly, from (1.5), (1.6), (1.7), (3.5). (3.8) and (3.9) itcan bededuced that fes.06/00dt=3a,Fin2)—6,~bmFo(02) =Sw(2)=6,(0). 1.10) LHANKEL-SCHWARTZ INTEGRAL TRANSFORMATION 1095 From Cauchy's theory ofresidues wefind the following integral representation of(3.9) bysot SO,69=ppj 2Fao)Fo")__by.) T piace Fo(w) twF,(0)+hFe(w)) Since §wires (om)dt=0°Fons(8) [ef.Gray etal, p.16,eqn. (25)), itcanbeinferred from (3.11) that j18S,(x,15A)dt=oojFrew) Fos()Gy,: Tybee Fu(w)(wFi(w)+hFy(w)} (3.12) Asanimmediate consequence of(3.11) and(3.12) wehave es IS@6MI<Gyamgoe—y 3.13) and (pao , a(4ynriGSaatRal<5S(4)aa (3.14) wherecsand¢,areconstants independent ofn,xandf. Next, itcanbeproved with anargument similar totheoneused inWatson'* (p.599)thatiff(1)isabsolutely summable intheinterval(a,b),0<a<b<1,then . {9S, (x,5A)£(O dt+0,a8n> 00 3.15) provided 0<x <1. Theorem 2—Let f(t) beafunction defined and absolutely summable inthe interval (0,1).Iff(t) isofbounded variation in(a,6)where 0<a <b <1,then theseries (3.1)converges tothesum#[f(x+0)+f(x—O)]atallpointsxsuch thata+A<x<b—4,A>0beingarbitrarily small. Proor:ByvirtureofTheorem 1theseriesZo,Fv(J,x)converges tothe sum}[f(x+0)+f(x—0)].Ourassertion follows directly from(3.10)and(3.15)topasstothelimitasn->co, 1096 J.M.MENDEZ Remark 1:Note that theinitial term byofourexpansion (3.1) only depends on thevalue ofA,whereas this term depends on A+ in the classical theory (cf.Watson", p,598),Moreover, therootsP,oftheequation 2F(z)+hFv(2)=0 arenotequal totheroots A,’s ofzJ,(z)+hJy(2)=(z)=0. 4. Tue Finrre HANKeL—SCHWaARTZ INTEGRAL TRANSFORMATION OF ‘THE SECOND KIND—APPLICATIONS According to(2.9) and (2.10), wedefine the finite Hankel-Schwartz integral transformation ofthesecond kind oforder v>—4bytheequation PovS00]=Fava)=fF, 0S(0de 4.0) whose kernel isthemodified Bessel function (2.6) and where f,,denote the roots of eqn. (2.7). The corresponding inversion formula is enPAFay(2)Fv(Pyx) -2 2,[FawOM=$00)=OteerSp —"=i(ah* +ap?—2vh)F4(P,a) 4.2) Theorem 2notonly guarantees existence of(4.1) but also ensures that inversion formula (4.2) holds. Ifweassume that thef€C*(0,a),f’(a)+hf(a)=0andh>0,weobtain themain operational formula ofthis transformation, i.e., . vd, : tw[2+BEE) ]=—6bw) 43) whatever thevaluesoff(0),andf’(0),provided theyarefinite. Iff' (a)+hf(a)#0 andh>0,weget - wei, too[rey+BEHy @]- Keo @+hs@ =PEhunFO. a) Remark2:Recallthatthefunction»y=F,(x)isasolution oftheequation Lymy +1+2y+y<0.ttsmultiplication byx"hasonlyrepercussions on thesign oftheparameter y,that is,thefunction y= FIQ= Fs) HANKEL~SCHWARTZ INTEGRAL TRANSFORMATION 1097 isasolution oftheequation trymy +By tym, Consequently, thesolution ofSturm-Liouville problem (L* +¥)dQ) =0 . , 2»)oy= NE@M=y@+ (a-*)$a)=0 is #2) =FF P(X) 4.5) where P,,P:,... denote thepositive zeros ofthe equation (2.7). The solutions (4.5) form aorthogonal system onthe interval (0,a)with respect totheweight function x'-4», Proceeding asbefore, wecannow introduce theintegral transform MLO =FLO=fo"FEONL) dx 4.6) whose inversion formula, inthecase h>0,isgiven by a 20FL)FEod A a=50)=SSsara'-*¥ (ah?+ap}—2vh)F$* (Pya) 4.7) Asimilar resulttothatproven inTheorem 2canbestated inrelation withthe convergence oftheseries (4.7), whenever v=—4. The main operational rule ofthetransform (46)is om[ros Spo |==6ae 48) Qv provided thatf’(a)+(—*)s@=0andh>0. Inthesequel weshall give afew examples toillustrate theuseoftheabove transformations insolving some important problems. (a) Let» beanyrealnumber. Wewish tofind thesolution oftheequation Ou w+ bee 1owgrtMEM efBmoo<r<ak>0) (49) satisfying theinitial condition u(r,0)=f(r) O©Srsa) 1098 3.M.MENDEZ andtheboundary conditions woo+hu(a,t) =0,foreveryv=0, or ued+(«-2)u@o =0,foreveryv=0. Byvirtue of(4.3) and(4.8), weconvert formally (4.9) into 2 kp -(2+et)ume=0, where hut (rsW), vz 0 FwO=Vas,(u(rDh¥S0. Hence, ketUn =Fiv(n) om (4.10) where Av (fl, +20 Fo) =VASO)» 50. Byapplying theinversion jformulas (4,2) and(4.7) to(4.10), wegettherequired solution if 2 ket fy ga PEFav (0)Fo(Par) emheh\sar> 20|" ES(att+ak)208)+FF(Pa) wnon4 SaPLFav(0)F5,(Py1)emheRE ay |ir [OO ca+or,+FI0.0 Note that when v~0the prablem (4.9) reduces tothe one considered by Sneddon" (eqns. 8-4-20, 33,34)onthediffusion equation, since inthis caseF,(x) =Jy(x)and p,=&arethe roots ofxJ;(ax)+hJp(ax)=0.Then,bothof formulas in(4.11) yield thesum solution andthiscoincides with theone achieved in the reference mentioned obove. Aprocedure similar totheone used byChurchill? (p.191), allows onetoesta- Dlish (4.11) asarigorous solution ofourproblem. HANKBL-SCHWARTZ INTEGRAL TRANSFORMATION 1099 Remark 3:Note that theequation (4.9)cannot besolved directly bymeans of thefiniteHankeltransformation, exceptwhenv=0,Nevertheless, thesimultaneous application ofthe figite Hank:I-Schwartz transformations (4.1) and (4.6) providesa simple method tosolve immediately theproblem (a), nomatter what the real value of ymay be. ()Many partial differential equations involving then-dimensional laplacian operator can also besolved byusing the transformation (4.1). Indeed, then-dimen- sional potential equation is GuGu, eu+o 0 (4.12)ext ax} oxi, where wu=u (XiyX35neyNeos 7) Ifwe seek solations which only depend on remxt+(x} +..+ x2)! andz,(4.12) reduces tothefrom (ef.Sneddon”, p.342) eupn2m|awat SR +Gee (4.13) Wefind now thesolution of(4.13) that satisfies theconditions 2.2)+hu(a2)=0(2>0,4>0) 4,0) =S0 (4.14) u(r, 2)+0,a82+©, bydirectly applying to(4.13) thefinite Hankel-Schwartz transformation ofthesecond kind oforderv=(n—3)/2.NowdenoteU,(z)=shaw(u(r,2).From(4.3)we seethat U,(2)satisfies theequation . Uy (2)_ —Ua)+TERE)oo whose solution is,inview ofconditions (4.14), U,(2)=F(n) e-tat where F(n) =daw fF) Again making useof(4.2), theformal solution oftheproblem posed byequations (4.13) 4.14)is Sy PEetal Fe(Par) Fn) 109- >——————. 2fA.15)Sar(ah?+ap)—2vh)$3(P,a) 1100 J.M,MENDEZ That (4.15) istruly asolution ofourproblem canbeproved assuming thatthe function f(r)issuchthattheabove series andtheseries obtained byapplyingthe operator Land2;converge adequately When v=O(i¢.,” =3)theproblem (4.13) consists offinding thebounded steady temperatures u(r,z)inthecylinder r€a,z >0,ifitisassumed that heat transfer intosurroundings attemperature zero takes place through thesurface r=a, according tothelinear law u,(a,z)=—hw(a,z). Remark 4:Theproblem (4.13) isusually solved bymeans ofthefinite Hankel transform only inthecasen=3(Colombo*, p.82).Now,bycombining thefinite transforms (4.1) and(4.6), itisfeasible tosolve this problem foreachn23,even more, foranarbitrary integer n. 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