mandal paper

Internat. J.Math. &Math. Sci. VOL. IINO.3(1988) 543-550543 ANOTEONBESSEL FUNCTION DUALINTEGRAL EQUATION WITHWEIGHT FUNCTION B.N.MANDAL Department ofMathematics University ofCalcutta 92Acharya Prafulla Chandra Road Calcutta 700009 INDIA (Received February 16,1987andinrevised formMay15,1987) ABSTRACT. Anelementary procedure based onSonine's integrals hasbeenused toreduce dualintegral equations withBessel functions ofdifferent orders askernels andan arbitrary weight function toaFredholm integral equation ofthesecond kind. The result obtained hereencompasses manyresults concerning dualintegral equations with Bessel functions askernels known intheliterature. KEYSWORDS ANDPHRASES. Dualintegral equations, Bessel function, Sonine's integrals, Hankel inversion, Fredholm integral equation. 1980AMSSUBJECT CLASSIFICATION CODE. 45F. I.INTRODUCTION. Inarecent paper Nasim [I]considered thefollowing class ofdualintegral equa- tions involving Bessel function offirst kindofdifferent orders askernels and arbitrary weight function It-2 J9(xt)[l+w(t)] #(t)dtf(x), 0<x<I, o It-28J(xt)#(t)dt-g(x), x>I. o Anoperational procedure based onexploiting theproperties ofMellin transform isused in[I]toreduce (I.i) toasingle integral equation involving Bessel function askernel. ByHankel inversion thiswasthenreduced toaFredholm integral equation ofsecond kind. However, thegeneral result in[I]wasobtained under theassumptions that n-81< +Iand E0,although thesecond assumption wasnotclearly stated. This isnecessary tomake %>-I(where %+-a+8)fortheHankel inversion tobevalid. Itisnotclear whether thisoperational procedure involving Mellin transform, canbeused ifthese restrictions ontheparameters p,9,,8 arenot satisfied. Thishasmotivated ustoreexamine thedualintegral equations (I.I) for arbitrary values oftheparameters p,9,a,8. 544 B.N.MANDAL Earlier Noble [2]used the'multiplying-factor method' based onSonine's integrals toaclass ofdualintegral equations withBessel functions ofsameorder ()askernels toaFredholm integral equation ofsecond kind. Schmeltzer and Lewin [3]alsoused thismultiplying-factor method toreduce another class ofdual integral equations involving first kindBessel functions ofdifferent orders anda particular weight function todualintegral equations with trigonometric sinefunction askernels inclosed formbythefunction-theoretic method. Later Ross [4]gave a simplified method ofconstruction ofexplicit solution totheclass ofdualintegral equations considered in[3]. AlsoRoseanddeHoog [5]considered another class of dualintegral equations with someparticular weight functions forexplicit solution. These dualintegral equations arose whenstandard integral transform approach wasused tosolve somemixed boundary value problems were solved byusing acomplex variable technique. Theexplicit solution tothedualintegral equations were thendeduced from thesolutions totheboundary value problems. Except forthese special class of dualintegral equations considered in[3,4,5], thegeneral class ofdualintegral equations (I.I) does notappear toadmit ofexplicit solution. References tomost of theworks ondifferent dualintegral equations withBessel function kernels canbe found in[1,2,4,6]. While in[2]dualintegral equations with first kindBessel functions ofthe sameorder askernels were considered, inthepresent paper aswell asin[I], integral equations withBessel functions ofdifferent orders askernels areconsidered. Hereweemploy themultiplying-factor method ofNoble [2]toreduce (I.I) toasingle integral equation withBessel function askernel. Invoking Hankel inversion, this single integral equation isthenreduced toaFredholm integral equation ofsecond kind. Theresult isobtained forallarbitrary values oftheparameters ,u,a,B. From thisresult manyknown results concerning dualintegral equations withBessel function kernels canbededuced. Asspecial cases, three sets ofvalues ofthepara- meters areconsidered. Thegeneral result obtained here thenreduces totheknown results forthese special cases. 2.REDUCTION TOAFREDHOLM INTEGRAL EQUATION OFSECOND KIND. Weassume that theparameters ,,a,B appearing in(I.I) aremost general. By using theformulae Idp()[x-J(xt)] (-I)px--ptpJ+p(xt), (2.1) dq x-q q(xt) (2.2) (x[xJ(xt) tJ_q where p,q arenonnegative integers, itisalways possible tochange theorders, to',' respectively suchthat'I,'>-3/2 (ofcourse, a,Barethen alsochanged) bychoosing psuitably. Thuswithout anyloss ofgenerality, wecan assume that theorders,oftheBessel functions in(I.I) satisfy therestrictions >-I, >-3/2. These restrictions arenecessary intheanalysis thatfollows. 2x2+IMultiplying each sideofthefirst equation in(i.I) by(r x where >-I,integrating from 0tor(<l) andusing Sonine's first integral intheform DUALINTEGRAL EQUATION WITHANWEIGHT FUNCTION 545 r I(r2-x2)x+IJ(xt)dx o 2i(+l)t--i+V+lrJu++l(rt),>-I, >-I, (2.3) weobtain ++I -2--I2F(+I) r ft Jk+l(rt)[l+w(t)] (t) dt o rx+l2f (r2-x) f(x) dx,0<r<I. (2.4) o Similarly, multiplying each side ofthesecond equation in(I.I) by(x2-r2)Dx-+I where >-I,integrating fromr(>l) to andusing Sonine's second integral inthe form I(x2-r2)x-+IJ(xt)dx r 2F(+I) t-n-Ir-+n+lJ--I(rt), -I < (2.5) weobtain 2rlr(n+l) r-+rl+lIt-2B-rl-1J (rt)(t) dt o --I I(x2_r2)nx-+l g(x) dx, <r<. r(2.6) Using (2.2) (with qreplaced by andxreplaced byr)in(2.4), weobtain ft-2a--l+EJv++l-(rt)[l+w(t)] (t) dt o ?r(+l)r(7 o2x2)(r f(x)dx, O<r<(2.7) where isanonnegative integer. Similarly using (2.I)(with preplaced bym and xreplaced byr)in(2.6) weobtain ft-28--I+mJ (rt)(t) dt-n-l+mo (-I)m2-nr-n-l+mm r(n+l) (3rf(x2-r2)n x-V+l o where misanonnegative integer. In(2.7) and(2.8) wenowequate thepowers oftandtheorders oftheBessel functions. Thisgives twoequations todetermine ,qasg(x) dx,r>I,(2.8) -2a- C+=-28 n+m u++ I-,=-n- l+m. Thus weobtain 546 B.N.MANDAL 2 u+aB+m2(2.9) Therequirements that and nmust begreater than-I(cf(2.3) and(2.5)) canbe satisfied bychoosing thenonnegative integers andmappropriately. Thus we obtain thesingle integral equation ft-J(rt)(t) dt 0 F(r) ft-J%(rt)w(t)(t) dt,0<r< 0 G(r) r>(2.10) where +v 2+ -9+ Y2(2.11) andF(r) andG(r) aregiven bytheright sides in(2.7) and(2.8) respectively. Nowinthesingle integral equation (2.10) werequire %>-IforHankel inver- sion. Ingeneral, thisrequirement maynotbesatisfied. Toovercome this difficulty, weuse(2.1) (with preplaced bysand xreplaced byr)in(2.10) toobtain ft-Y+sJ%+s(rt)(t)dt 0 Fl(r)-(-l)Soft-Y+sJ+s(rt)w(t)(t)dt, 0<r< G(r) r>(2.12) where Fl(r)(-l)Srl+Srs([r-)"F(r)], Gl(r)(-l)Srl+Srs([r-G(r)],(2.13) and sisanonnegative integer. Whatever bea,B wecanalways choose %+s>-I. Then theHankel inversion canbeinvoked in(2.12) togive aFredholm integral equa- tion ofsecond kind in(x) as (x) xY-s+lorFl(r)Jx+s(rX)dr+frl(r)J)t+s(rX)dr -y+suJ%+s+l(u) J%+s(X) xJ%+s+(x)J%+s(U)fu w(u)(u)2 2du O U X(2.14) DUALINTEGRAL EQUATION WITHANWEIGHT FUNCTION 547 where I,aregiven in(2.11) and sisanonnegative integer tobechosen appropriately soastomake t+s>-1. 3.SPECIAL CASES. 3.1., -a/2, B0where >max(-I, a)and 0<a<2.Thedual integral equations arethen fta[l+w(t)] (t)J(xt)dtf(x), 0<x<I, o (xt)dtg(x) x> f(t)Jo(3.1) These dualintegral equations were considered byNoble [2]. Theconditions >-I and >-3/2 aresatisfied automatically. Nowfrom (2.9) a a++,,n=-I-+m where E,marenonnegative integers tobechosen such that >-I, >-I.We a achoose O,m sothat -I+>-Iand->-i. Also a a% +>-I, y-.Hence wetake s0in(2.14). Then af 2_u2)_l+a/221-a/2-v- u+I(r f(u) du,Fl(r)F(r)r(a/2)r o 2a/2 _+a Gl(r)G(r)r(l-a/2)r-Id$(u2_r2)-a/2l-v d-ug(u) du, o sothat theFredholm integral equation in(x) is (x)(2x)l-a/2rl_V_a/2 (rF(a/2)f fru+l2u2)-l+a/2 o of(u)du}J+a/2(rx)dr l-a/22a/2 -+a/2d2)-a/2I- xrdrf(u2-r ug(u) du}"J+a/2(rx)dxr(l-a/2)r xl-a/2Iua/2 ouJ+a/2+l(u)J+a/2(x)-xJ+a/2+l (x)Jv+a/2(u) w(u)(u)2 2du u x(3.2) Thisresult wasobtained in[2](cf.eq.(4.7)). Putting g(x) O,w(u) O,weobtain theexplicit solution ofthedualintegral equations fta(t)J(xt)dtf(x), 0<x<I, o f(t)J(xt)dt O,x> o where >max(-l,-a)and 0<a<2,as (x)(2x)l-a/2 r(a/2)Ira/2+l(rx){ fv+I a/2-1J+a/2f(rv)l-v2 o o(3.3) dv}dr. (3.4) 548 B.N.MANDAL This coincides with theresult given in[7]. 3.2.I-81< + and 0. This casewasconsidered in[I]. Then >-i, andwemaymention here that therestriction 0wasnotstated explicitly in [I]. Unless thisrestriction holds good, itisnotpossible touseHankel inversion with theorder oftheBessel function as.Tomake >-I,n>-I,wechoose I,m in(2.9). Also as >-I,wechoose s0in(2.14) sothat the dualintegral equations (2.1) inthiscasereduce to #(x)xY+lE2-Y+2s -y+2-rug+l2u2)-2 Ir Jl(rx)dI (r f(u)du}r(-2+l) o o 2-Y+28IrAJ%(rx)dI(u2-r2)Y-28uI-Mg(u)du}F(Y-28+l)r uJl+l(U) Jl(x) xJl+l(x)J%(u)Iu-Yw(u) #(u)2 2du o u x(3.5) Thisresult does notseem tocoincide with thegeneral result given in[I]. However, theknown results forthefollowing twospecial cases given in[6]aswell asin[I] arededuced from (3.5) implying that (3.5) isalso correct. When 90, 1/2, 80,(2.11) gives =-1/2 and y.Substitut- ingin(3.5) weobtain after simplification r #(x)2__x/cosrxdIuf(u) o o(r2_u2)I/2du +2__xICOSrx Iu5(u) o o(u2_r2)I/2dudr xfw(u)#(u) sln(u+x)+sln(u-x)duw u u+x u-xo(3.6) Thisresult canalsobededuced directly from(2.14) bychoosing s0, I,m0. When O, -I/2, 80,theintegral equation for#(x) cansimilarly bededuced from (3.5). Inthiscase I/2,y-I/2. After simplification we obtain r #(x)--2IsinrxIuf(u) o o(r2_u2)1/2dudr 2IsinrxIu(u) r(u2_r2)112du -1fw(u)#(u)sin(u-x)- sin(u+x)dxw u-x u+xo(3.7) (3.7) canalsobededuced directly from(2.14) bychoosing s0, O,mI. 4.DISCUSSION. Theoperational procedure based onMellin transform toreduce theintegral equa- tions (I.I) toaFredholm integral equation ofsecond kind asgiven in[I]seems too DUALINTEGRAL EQUATION WITHANWEIGHT FUNCTION 549 complicated andlengthy compared toNoble's multlplying-factor method based onSonlne's integrals asdemonstrated inthis note. Further, themethod outlined here canbeused toattack thedualintegral equations (I.I) forarbitrary values oftheparameters 9,,=,8 andtheweight function. ACKNOWLEDGEMENT. Thisworkwassupported inpartbytheUGCproject No.F.8-8/85 (SRIII). REFERENCES I.NASIM, C.Ondualintegral equations withHankel kernel andarbitrary weight function, Internat. J.Math. andMath. Sci. 9(1986), 293-300. 2.NOBLE, D.Thesolution ofBessel function dualintegral equations byamultiply- ing-factor method, Proc. Camb. Phil. Soc.59(1963), 351-362. 3.SCHMELTZER, R.A.andLEWIN, M.Function-theoretic solution toaclass ofdual integral equations andapplication todiffraction theory, Quart. Appl. Math. 21(1964), 259-283. 4.ROSS, R.A. Anintegral equation arising indiffraction theory, Wavephenomenon: modern theory andapplication, North Holland Math. Stud. 97(1983), 453-459. 5.ROSE, R.F.andDEHOOG, F.R. Exact solutions ofcertain dualintegral equations andtheir asymptotic properties, q.Jl.Mech. AppI. Math. 36(1983), 420-436. 6.SNEDDON, I.N. Mixedbounar value problems inpotential theory, JohnWiley and Sons, NewYork, 1966. 7.TITCHMSH, E.C. Introduction tothetheory ofFourier integrals, Oxford Univer- sityPress, 1937.