nassim dual integral

Indian J.pare appl. Math. (2): 323-340, March 1986 ON SOME DUAL INTEGRAL EQUATIONS C.Nasim AnD B.D.AGGARWALA Department ofMathematics andStatistics, University ofCalgary, Calgary Alberta, Canada T2NiNA (Received 26September 1983) Wedevelop anelementary method tosolve thedual integral equations of the type Fnangea apo,oce<s ihyOO)OD)dt=BX),x<1 where hyand hyare Bessel functions ofthe fist and the second kind. First weconsider thedual equations with ¢(2)=0 and determine anappropriate form oftheunknown function¢itermsofanarbitraryfonction.‘Thea this arbitrary function isevaluated sothat stistes theequations. Simi. larly thesecond setofdual equation with /(#)=0 issolved and combining the solutions ofthese two sets the solution oftheorignal setofdual equttions isobained 1. INTRODUCTION Inthis paper wedevelop anelementary method for solving certain dual integral equations ofthetype [hans@a so, o<x<! \ et) 2 i Jmans@a=e@, 1cxc| :J The method consists ofdecomposing thesystem ofeqns. (1.1) into two setsofdual integral equations. Wederive the solution ofthe first set, which isthesystem of eqns. (1.1) with g(x)=0,and then find thesolution ofthesecond setofequations, 304 ©.NASIM. AND B.D. AGGARWALA thesystem (1.1) with f(x) =0. Combining the solutions ofthese two sets gives usthesolution of(1.1). Tosolve each ofthe two above mentioned sets ofintegral equations, animportant part ofour technique istoestablish asuitable form of thefunction ¢,interms ofanappropriately defined function. We then find thevalueofthisunknown function,sothat¢satisfiesboththeintegralequations ofthat set, thus establishing itssolution, Wemake extensive useofthe ErdelyiKober operators and their properties. This type ofapproach forsolving dual integral equations has been used widely ina variety ofmethods (Gordon 1954, Noble 1958), though notexpressed inthesame notations inallthe cases. Agood account ofsuch methods can befound in Erdelyi and Sneddon (1962) and inSneddon (1966, pp. 84-96). To illustrate our technique, we shall first obtain solutions ofsome simplepairsofdual integral equations. Then weshall apply the procedure tosolve dual integral equations ofTitchmarsh’s type and integral equations whose Kernels involve Y-and H-functions, where ¥isthe usual Bessel function ofsecond kind and HisStruve'sfunction. Asistheusuallythecase,theanalysisusedinsolvingthe dual integral equations inthis paper ispurely formal 2. SoMe SIMPLE PAIRS OFDUAL INTEGRAL EQUATIONS (i) We first consider thedual integral equations ofBeltrami's type, J4)bt)dr=f(x), O<x<1 .Q.1a) Js@ncnd=o, 1<x<0 1B) Which arise intheproblem oftheelectrified disk. Weshall write (2.1b) as J86 400antdi=«@)HO-9) where 0<x <09,a(x)anarbitrary function and Hbeing theHeaviside function. ‘Then bytheHankel inversion, rgn= j&(x)Jp(xt)(xt)de. DUAL INTEGRAL EQUATIONS 325 Ifwelet(Erdelyi efal.1954a, p.11) 2f oxy Jocory= 2|EOD ay then eget ifa(x)(xt)deJBSa 2,|omona|TeoSs 60)=|200s(yo 122) where pw=2 Jae Tofind anappropriate value of (w), so.that¢satisfies(21a)aswell,substitute ¢in(2.12) toobtain so=|Just)dt|8)cos(ut)Ox<I free [cos(uyJy(xta =j—2)__day,(Erdelyietal.1954a,p.43) )vee whence (Sneddon 1966, p.41) 24f x -24 as.sm-2 Z|TEax 326 ©.NASIM. AND B,D. AGGARWALA Thus from (2.2) _2,f 4(¥O 4smn2eJcoronat|tee which now satisfies both (2.1a) and (2.1), giving usthe solution ofthesystem (2.1). Amore familiar form ofthis solution isobtained byintegrating w-integral byparts, and simplifying, that is 2(¥@ {vdj t=2rgeose{ LOdn1fPR|usinur)-£0dy $(0)=eosjFee +|eeJoomronan Iff(x)=1.0 <x<1.thesolution ofthe corresponding dual integral eqns. (2.1), becomes sc= sine ‘another well-known solution ofthis particular case. Gi) Let i$(1)cos(xt)dt=f(x), O<x<1| °{ t(2.3) Jse@sinand=2@, ters-| ‘The dual integral equations ofthistype,withtrigonometrical kernels, arisefre- quently intheanalysis ofmixed boundary value problems intheplane; such ascrack problems inthetwo-dimensional theory ofelasticity. Weshall consider two cases of(2.3). Case a— First consider j$(1)cos(xt)d=f(x), O<x< 1 (2.48) J(0)sin(xt)dt=0, 1<x<ce, ve(2.4b), DUAL INTEGRAL EQUATIONS 327 Equation (2.4b) can bewritten as (t) sin(xt) dt=(x) H(I—x), O<x< forsome a(x),Hbeing theHeaviside function. Bytheusual sine-transformation $=2j(3)sin(x1)de Now ifwelet,(Erdelyi etal.1954b, p.7) sin(xt)=Jadu ‘Then 2 separ |Meu ewVe1|dsJGeo.ty 2,f few, =t|udy(ut)du{2a, v=J.“[a * or =1|Bender de 23) where w=9/5Jaes Noweshall write (2.4a) as af. soyadjPgMsinQxd, O<xeL 328 ©.NASIMANDB.D.AGGARWALA Substituting for¢from (2.5), weobtain. 7)alsin(xt)dtj8(w)Jo(ut)du at 2-Zj8(u)duJsin(xt)Jy(ut)dt 4{_3w x&JVeoMeOse! due to(Erdelyi 1954, p.10)whence (Sneddon 1966, p.41) w=2uf £2 Bw)==“Jeate Hence from eqn. (2.5) zt sf) 0. é—2 A JQ) ax (2.6) w=2aWJ(ut)ctJSes Which gives usthesolution oftheintegral equations (2.4). Case b— Let J¢(t)cos(xt)dr =0,OCx<l ves(2.Ta) [s@sin@n dey, Lex<o (2.70) ‘Once again, from (2.7a), using thecosine transformation 3% simJzx(x)cos(xt)dt foranappropriately defined function«(x).Weusetherepresentation (Erdelyietal.1954b,p.7) DUAL INTEGRAL EQUATIONS 329 at)=f Helut) cos (xt)=1JaStu inthe above expression and change theorder ofintegration toobtain oat YBateturdu 2.8) where =f2f © 8)V2|75sdx ‘Now eqn. (2.7b) isequivalent to e@e- fj118(eos(xt)dt<x<oo and substituting for¢from (2.8), get y=—Zlcostar |p09Je ==H|ede[cos(x0)Jet) 4° sw yee, ~&|phe tex Then byinversion (Sneddon 1966, p.42) Hence, from (2.8) -2,, feso=Fe]wttede|reae (2.9) 330 C.NASIM.ANDB,D.AGGARWALA gives thesolution ofeqns. (27). Now combining (2.6) and (2.9), weobtain 2,f (fae=fwaytuaufLOrde sm=2 JtonaeTRS 2j ffac)Z| wdgiuydy|ESae 42.10) aeA rs 2.10) which isthesolution ofthedual integral equations (2.3). ‘The solutions ofavariety ofdual integral equations with trigonometrical kernels can bededuced from the solution (2.10); weshall mention only one important case below. Write (2.4a) as j1-99(0)sin(xt)dt=ifide, O<x<1 =F@, 0<x<1 then thepair ofintegral equations (2.4) become J1g(sin(xt)dt=F(x), O<x<1 [#esingayd=0,Lexce. The solution ofthispair ofequations isthen derived from (2.6) togive (Sneddon. 1966, p.106) ey=? rf4Jy(ut)u|SEQdx. | }Vee IfweputF(x) =x,then 4 =40) giving thesolution ofthedual integral equations DUAL INTEGRAL EQUATIONS 331 J16(sin (xt)drx,O<x<1 if$(t)sin(xt)dt=0, Loxces 3. Duat INTEGRAL EQUATIONS OFTITCHMARSH TYPE Consider ibt d=f, O<xel Jres@nendi~69,<x<on G1) Case a—Let if1g()J,(xt)dt=f(x), O<x<1 (3,2a) Jr®s@mond=o, 1<x<e (3.26) Write (3.2b) as J8-084(1)Jy(xt)(xt)?dt=2)H(I-x), v>4 and byHankel inversion. PIA) =ja(x)Jv(xt)(xtde,v>—b. Now using expression Jot) =Po weesf 8(taut Jy(ut)du lv reap wt(tuys Is(ut) 322 €.NASI AND B.D, AGGARWALA where 1=4p-+4v— +48 and —1<1<y, inthe above integral and thenchanging theorderofintegration weobtain raegmeee reromn |atedyuy To-y)}* xJBYw(x)(atutdx $0)= |Bla)Juut) 3) say. Substituting in(3.2a), so=f tJ,OxdtiB(u)Ja(ut)du,OC<I -fewdujPJ,(xt)Jn(ut)dt ef paaratudnOe<I er aanOC) ,Onx< where newt oath 1<4<pandlacy. 2.4) or, LO) =27 Tassapspen BAM BD] using Erdelyi-Kober fractional integral operator 7,defined inSneddon (1966, p.48). Then, bytheusual inversion BG) =PM an ee (10) = HN Hrapran LO] DUAL INTEGRAL EQUATIONS 333 Hence from (3.3) 8(0=28Brn JWMOJy(Ut)IapsenLf)de, G5) =4.(0 which isthesolution ofthedual integral equations (3.2). When theconditions (3.4) arenotsatisfied, (3.5) stillgives asolution ofthesystem (3.2), buttheoperator Ihas tobeinterpreted appropriately. Case b— Let J46()I,Gd=0,OS<1 B.C) JsOrGnd a2, 1<xc3o 3.66) Asbefore, bytheHankel inversion andforanappropriate «(x),(3.6a) gives eminem |seyJ(Grd, >=> Ifweusetherepresentation (Erdelyi eral.1954b, p.25). I(t)=Qo eeJw="(utx2)Jy(ut)du a ro) } where 1=da+iy~2+B and »<< 24+ 3/2, inthe above integral and change theorder ofintegration, wethen obtain: $()=2paerenfeJa(ut)duf(x)he ro" J 2 x(ut=x8hPF dye =tetera JBu)Ju(ut)du,say. QB.) 334 C.NASIM AND B.D. AGGARWALA Substituting in(3.6b), stays|restemre acondtfFplunder diex>1 =|8(u)duf178J,(et)Jn(ut)dt =bac)” JBu)we de where —1<v<qand ¢<4 <2+3/2. Orusing theErdelyi-Kober operator K(Sneddon 1966, p.50) B08) =2" Kyrays ay[XB I] which oninverting gives B(x) =2 Keyore lgC9 Hence from (3.7), (1) Deeer FWY, (ut)Kaaygvoren [8] 2.8) =a) giving asolution ofthesystem (3,6). Now combiaing (3.5) and (3.8), wehave $O= HO FEO G9) thesolution ofthedual integral equations (3.1) ofTitchmarsh type. Inparticular if4—»>0and v—4>O,ic. «~¥<2(3—2),then(3.9) becomes =PG) }* Qwn r@ 4oopavesfaeryet)dx{tate aes ro=%) ;+(xifwxTYe(a DUAL INTEGRAL EQUATIONS. 335 Also if«=y,then $(0)=eBeB{jP28)Jeoa(xt)IamswaLf) +ixtBOIta(xt)Kriapaons 0-8[8]ax} G.10) isthesolution ofthesystem i8(Ip(xdtf@),O-xe1I . it 6.) ¢ ii184b(0)J,(xt)dt=g(x), 1<x<0. . Jj Further, if0<p—«<1,then(3.10)gives ra 87 $0)gesBTFe geeGtdy xjwhet(xtuayPa f(u)ds 28eeejao rl -—pase5'}”Spston(x1)de xifuP(ue=xtg(u)du @well known solution ofthesystem (3.11). 4. Duav INTEGRAL EQuations INVOLVING Y-FUNCTIONS First, wedefine operators /a,x(a,x:2)and Ka,» (x,6:2) which aresimple generalization ofErdelyi-Kober operators, bythefollowing equations. 336 €.NASIM AND B,D,AGGARWATA irsasarof(tryed emf(Ddt,a>0 Isa (ax: DIS =4 ware&)(tryef(1)dt, t : ~i<e<0 and f e \rayal(Paxtpetete(Qdt,a>0 Kaa(x,6:2)(f=< . jot od 2xtyeites(-Fae£feamepa,l : ~I<a<0 Note that Inn (0,822) =Ina and Kaya (X,022)=Kaya. Itcan beeasily shown that Te, (0,822) =Inyayea (a,X2) and KG, (6)8:2)=Kegasen (0,6:2) defining theinverse operators. Now consider thedual integral equations Jr™6()Y, (xd=f), O<x<1 4.) [res@nand =e, 1<x<0 where 2(«—8)=v—4,|#| <4, andlv|<4. Case a— Let JPgO%Od=SQ), 0<x<1 (4.28) DUAL INTEGRAL EQUATIONS 337 i18g(1)Ye(xt)dt=0.Lox o, (4.2) We write (4.2b) as j1-20-10 (8)Vy(x)(xt)!dt=@(x)H(=x) and bytheusual inversion, 818 6m[0Houta diy||<4 4.3) where HyisStruve’s function oforder v.Now substituting thevalue of¢from (4.3) into (4.2a) and simplifying, weobtain f(x)=j(uae ufpewY,(xt)Hy(ut)dt, O<x<1 -{2(u)widujte-¥*1Y,(xt)He(ut)dt since 2(2 —§) =v —y. The tintegral can beevaluated (Erdelyi etal.1954b. p.114) togive f=2rataQ)wtstaxtrtdu,OSxT row©J a where y—» >0. Or, S02 Kiraprven Ge152) UMH a) whence, 2(2)=2PMKecrapepey 1DUS] Substituting in(4.3, gives usthesolution 8(0)=26rphHjhetHy(ut)Ke-arapop-v (uy1:2)Efe.4.8) ofthesystem(4.2).Ify—<0,thentheoperator Kistobeinterpreted accordingly. Caseb—Let ifPrg(DY,(d=0,O<x<1 (45a) 338 ©.NASIM AND B,D.AGGARWALA ifPE)KONdrg(x),1<x<0o (4.50) where 2(@ —3)=¥—4,|¥] <#and| |<+.Once again weinvert (4.5a), foranappropriately defined function a(x),toget eae ia(u)Hy(ut)Whdu,fw|<4. al.6) Substituting in(4.5b) and simplifying, weobtain ea)=feapurds [mmKonH,(udex>t =rG@e)ala tax dy,oy>0 whence (x) =DK, pve [8 Then from (4.6) (are ree[eer Cut)Kyron Le) (A.D) giving thesolution ofthe dual integral equations (4.5). This isalso thesolution even ify—p> 0. Now combining (4.4) and (4.7), wehave $mDereb[UHH (Ut)Keeper te1:DFT sparen [8 Hy(ut)Kysela (48) thesolution ofthe system ofdual integral equations (4.1). Inparticular if0<v —p< 1,then (4.8) becomes DUAL INTEGRAL EQUATIONS 339 FXO)rte'\«He(ut)duFe x|Gt=wytrat”f(a)de 2e-w° f =2 fae ot—usyretxg(x)de, +pty|wo"Hy(ut)du|(2—uryerrtx-¥8(x) Alsoif—1<¥=<0,then awe7 ( =ETpe(ey Bm(xtuPe-¥9f(x)dh $=Fors" J»Hey(ut)du[28-*(a8uberf(3)dc ae tathf i" a +risa tieA,(ut)du xifate(xtuaeg(x)de Ieweletf(x) =x*andg(x)=0,then $= HaO, ly—wl<t giving thesolution ofthesystem (4.2). 5. Duat INTRGRAL EQuaTIONS wiITH H-FUNCTIONS Consider Jrb(OH,(xt)dt=f), O<x<l j194(9)Hy(xt)dt=f), 1<x<0o 25:1) where2(8—a)=v—w,|v]<}and|p|<d. For deriving the solution ofthis system, we proceed exactly asbefore and establish first thesolution ofthesystem with g(x)=0,which is 340 (C.NASIM ANDB,D,AGGARWALA 6(0)=ee jwrt¥y(ut)IypapeveyLf]de BAM where /istheErdelyi-Kober operator. Then weletf(x) =0in(5.1) and find the solution ofthesystem, whichis seyrerees [wey aIAEE? LP =4.00) where 7(1, x:2)isthegeneralized Erdelyi-Kober operator. Hence 4O= HOF HO isthesolution ofthedual integral equations (5.1) REFERENCES: Erdelyi.A.eral.(19548).TablesofIntegralTransforms, Vol.I.BatemanManuscriptProject McGraw-Hill Book Co, Inc, New York ———=(19540), Tables ofIntegral Transforms, Vol. I. Bateman Manuscript Project MeGraw-Hill Book Co., Ine., New York Erdelyi,A.,andSneddon,I,(1962).Fractionalintegrationanddualintegra!equations Canad. J.Math. 14,68. Gordon, A...(1954).Dualintegralequations.J.LondMath.Soc,29,360. Noble,B.(1958).Certaindualintegralequations.J.Math.Phys.,37,128 Sneddon,I.N.(9166).MixedboundaryvalueproblemsinPotentialTheory.NorthHollandPubl. Co., Amsterdam