Transmutation Method for Solving Erdelyi-Kober Fractional Differintegral Equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 211, 347᎐364Ž. 1997 ARTICLE NO. AY975469

Transmutation Method for Solving Erdelyi´ ᎐Kober Fractional Differintegral Equations

Virginia S. Kiryakova

Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, 1090, Bulgaria, and Istituto per la Ricerca di Base, Monteroduni, Italy

and

Bader N. Al-Saqabi

Department of Mathematics and Computer Science, Faculty of Sciences, Kuwait Uni¨ersity, P.O. Box 5969, Safat, 13060, Kuwait

Submitted by H. M. Sri¨asta¨a

Received August 5, 1996

By means of fractional calculus techniques we find explicit solutions of Volterra integral equations of the second kind and fractional differential and differintegral equations, involving Erdelyi´ ᎐Kober fractional integrals and derivatives. Also, some hypergeometric integral equations have been considered and solved as double Erdelyi´ ᎐Kober equations of the second kind. We use the transmutation method to reduce the solutions of all these equations to known solutions of simplerŽ Rie- mann᎐Liouville. equations of the same type. Some examples are given. ᮊ 1997 Academic Press

1. INTRODUCTION

Fractional integral, differential, and differintegral equationsŽ FIE, FDE, FDIE.Ž. , involving the Riemann᎐Liou¨ille R-L integrals of order ␦ ) 0,

␦ 1 x x ␦ ␦y1 1 ␦y1 RyxŽ.s HH Žxytytdt . Ž. s Ž.Ž.1y␴ yx␴ d␴, ⌫Ž.␦00⌫Ž.␦ Ž.1

347

0022-247Xr97 $25.00 Copyright ᮊ 1997 by Academic Press All rights of reproduction in any form reserved. 348 KIRYAKOVA AND AL-SAQABI

␦ ␦ and R-L deri¨ati¨es Ry [ D , ␦ ) 0, d n ¡ ␦yn ␦ RyxŽ.,if␦noninteger, n s wx␦ q 1 DyxŽ.s~ž/dx Ž.2 Žn. ¢yxŽ.,if␦sninteger, have been recently solved explicitly by various authors inw 4, 5, 7, 10, ␦ 13᎐16, 18᎐20, 22, 23x . The solutions of the first kind equations RyxŽ.s fxŽ.are well known. Abel Ž 1823 . was the first to solve effectively such an equation with ␦ s 1r2Ž. called now the Abel integral equation by means of fractional calculus, thus giving a good motivation for further development of this topic.

DEFINITION. The equations

x yxŽ.y␭HKx Ž,tytdt .Ž.sfx Ž.,3Ž. a where fxŽ.,Kx Ž,t .are given functions, ␭ is a parameter, and yxŽ.is the sought solution, are called Volterra integral equations of the second kind. Solutions of the R-L fractional integral equation of the second kind

␦ ˜˜yxŽ.y␭Ryx Ž.sfx˜ Ž. Ž.4 have been found respectively by Hille and Tamarkinwx 7 by means of the Laplace transform, Ross and Sachdevawx 16 by the techniques of fractional calculus, and many others by means of operational calculusŽ see, e.g., Gorenflo and Luchkowx 4. :

x ␦y1 ␦ ˜yxŽ.Ž.Ž.sfx˜˜q␭HxytE␦,␦␭ Ž.Ž.Ž.xytftdt.5 0 Analogously, the Cauchy problem for the R-L fractional differential equation

␦ Dyx˜˜Ž.y␭yx Ž.sfx˜ Ž. Ž.6 ␦yj ½Dyx˜Ž.xs0sbj, js1,2,..., n; n y 1 - ␦ F n has a solutionŽ see Examples 42.1, 42.2 in Samko, Kilbas, and Marichev wx18. :

n x ␦yj ␭ ␦ ␦y1 ␭ ␦ ˜ ˜yxŽ.sÝbxj E␦,1q␦yjŽx .qH ŽxytE .␦,␦ Žxytftdt . Ž. . js1 0 Ž.7 FRACTIONAL DIFFERINTEGRAL EQUATIONS 349

Recently, Al-Saqabiwx 19 and Tuan and Al-Saqabi wx 23 , using techniques similar to those inwx 16 , have found solutions to more general differintegral equations involving both R-L fractional integrals and derivatives,

␮ ␯ Dyx˜˜Ž.y␭Ry Ž. x sfx˜ Ž., ␮)0, ␯ ) 0,Ž. 8 namely,

ny1 ␣ ␮yjy1 ␭␮q␯ ˜yxŽ.sÝjxE␮q␯,␮ykŽ.x js0

x ␮ ␯ ␮y1 ␭ q ˜ qHŽ.xytE␮q␯,␮ Ž.xytftdt Ž.Ž..9 0 All of the above solutions involve the Mittag᎐Leffler Ž.M-L functions Žfor properties and applications see, e.g.,wx 2, 3, 10, 14 . ϱ x k Ex␣,␤Ž.sÝ ,␣)0, ␤ ) 0.Ž. 10 ⌫Ž.␣k ␤ ks0 q In this paper we consider integral, differential, and differintegral equations involving more general operators of fractional integration and differ- entiation, called Erdelyi´ ᎐Kober Ž.E-K fractional integrals and deri¨ati¨es, respectivelyŽ seewx 21, 10. ,

␥ , ␦ yŽ␥q␦ . ␦␥ 1r␤ IyxŽ. xRxyxŽ. ␤ ␤ s xªx ␤Ž␥ ␦. xy q x ␤␤␦y1 ␤␥␤ s HŽ.Ž.Ž.xyttytdt ⌫Ž.␦0 1 1 ␦y1 ␥ 1␤ s HŽ.1y␴␴yx Ž␴r .d␴ Ž.11 ⌫Ž.␦0 and

␥ , ␦ y␥␦␥q␦ 1r␤ Dyx␤Ž.sxDxyxŽ. ␤,12Ž. xªx with real ␦ ) 0, ␥, and ␤ ) 0. Evidently, for ␥ s 0, ␤ s 1,Ž.Ž. 11 , 12 turn intoŽ. 1 , Ž. 2 . The additional parameters ␥, ␤ allow more generality and these operators have found a large number of applications in analysis, mathematical physics, and other disciplineswx 21 . Fractional differintegral equations and Volterra type integral equations of the second kind, in¨ol¨ing E-K operators Ž.Ž.11 , 12 ,

y␤␦ ␣, ␦ ␤␯ ␮,␯ x D␤ yxŽ.y␭x I␤ yxŽ.sfx Ž.,13Ž.

␤␦ ␣q␦,␦ ␤␯ ␣y␯,␯ yxŽ.y␭Ž.Ž.x I␤x I␤yxŽ.sfx Ž., ␦)0, ␯ ) 0,Ž. 14 350 KIRYAKOVA AND AL-SAQABI

arise very often in various problems, especially describing physical pro- cesses with aftereffects. However, solutions of Ž.Ž.13 , 14 ha¨e not been found explicitly by now. To solve each of the above equations, we apply the transmutation method. It consists in applying suitable transformation operators that allow us to find solutions of new and more complicated problemsŽŽ.Ž.. like 13 , 14 via their ‘‘translation’’ to known solutions of simpler old problemsŽ in our caseᎏto solutionsŽ.Ž.Ž. 5 , 7 , 9 of Eqs. Ž.Ž.Ž. 4 , 6 , 8 with R-L operators . , see, e.g.,wx 6; 10, Definition 3.5.1 . For the simplest cases ofŽ. 13 an operational method based on a Laplace type transformationŽ the Borel᎐Dzrbashjan integral transform, see Dimovski and Kiryakovawx 1 , Kiryakova w 10, Chap. 2 x. is also applicable, similar to that inwx 7 . Then the solutions are expressed in terms of the convolutions of E-K operators, found by Kiryakovawx 9, 10 . Suitable substitution could also reduce Eq.Ž. 13 with either ␦ s 0or␯s0 to R-L equationsŽ. 4 , Ž. 6 . However, we apply the transmutation method that turns out to be more effective in the general casesŽ.Ž. 13 , 14 and gives an idea how Eq.Ž. 3 with more complicated kernels Kx Ž,t . can be attacked. The transmutation operators used in this paper are fractional calculus operators. We look for solutions in spaces of weighted continuous functions of the form Žk. p ˜˜Žk. C␮[Ä4fxŽ.sxfx Ž.;p)␮,fgC w0,ϱ., Ž0. C␮[C␮, with real ␮,1Ž.5 ␮ although spaces Lpp, L can be considered too.

2. ERDELYI´᎐KOBER INTEGRAL AND DIFFERENTIAL EQUATIONS

By analogy withŽ. 4 , Ž. 6 , inwx 20 we consider integral and differential equations of formŽ. 13 , involving either an E-K fractional integral Ž. 11 or an E-K fractional derivativeŽ. 12 with arbitrary parameters, i.e., first we restrict to the case when either ␦ s 0or␯s0. The following explicit solutions have been found inwx 20 , summarized in Theorems 1, 2. Ž. Ä4 THEOREM 1. The unique solution y x g C␤␮ , ␮ G max 0, y␥ y 1 of the Volterra integral equation of the second kind Ž.15 , i.e.,

␤␤␦y1 xŽ.xt ␤␦␥,␦ y␤␥ y ␤␥ ␤ yxŽ.y␭xI␤ yxŽ.syx Ž.y␭xtH yŽ.Žtdt . 0 ⌫Ž.␦ sfxŽ.,16Ž. FRACTIONAL DIFFERINTEGRAL EQUATIONS 351

Ä4 with f g C␤␮ , ␮ G max 0, y␥ y 1, has the explicit form of a con¨olutional type integral:

x ␦ 1 ␦ y␤␥␤␤y ␤␤ ␤␥␤ yxŽ.sfx Ž.q␭xxH ŽytE .␦,␦ ␭ Žxyttftdt . Ž.Ž .. 0 Ž.17

Proof. The homogeneous equationŽ.Ž 16 f ' 0 . has only the trivial solution y ' 0 and this yields the uniqueness of the solution g C␤␮ in the nonhomogeneous case. To apply the transmutation method to Eqs.Ž.Ž 4 , 16 . , that is, to transform the former into the latter, we need transformation relating the operators

␤␦ ␥ , ␦ ␦ ␥ , ␦ ␦ ␦ 0, ␦ L␦, ␤␤[ x I , L ␦,1[ x I 1 , R s x I1 .

They are given by the E-K fractional integral

␥y1 x Ž.x ␶ 0,␥ y␥ y TsI1 in the form T ⌽Ž.x [ x H ⌽Ž.␶ d␶ Ž18 . 0 ⌫Ž.␥ and by the mapping

y1 y1 ␤ ⍀ : C␮¬ C␤␮ , ⍀ : fxŽ.¬fx Ž .,␤)0.Ž. 19

Let us have in mind that E-K fractional integralsŽ. 11 preserve spacesŽ. 15 Žseewx 10, Chap. 1. , in a sense that

␥ , ␦ Žn. I␤: C␤␮¬ C ␤␮; C ␤␮ , for ␮ GyŽ.␥q1,ny1-␦Fn and whence all the operators, involved in our considerations, ␦ y1 T, R , L␦,1, L␦,␤␮, ⍀ act in C or C␮␤with either ␮ Gy1or␮Gy␥y 1. To encompass these spaces in both cases ␥ G 0 and ␥ F 0, we use Ä4 further the denotation C␮, C␤␮, ␮ G max 0, y␥ y 1. The techniques of fractional calculuswx 10, Chap. 1 allow us easily to Ä4 establish the following similarity relations in C␮, ␮ G max 0, y␥ y 1:

␦ ␦ ␦ 0, ␦ ␦ ␥, ␦ TR s LT␦,1 , i.e., T : R s x I1 ¬L␦,1sx I 1 , ␥gR,20Ž.

y1 y1 y1 ⍀ L␦,1 sL␦,␤␦⍀ , i.e., ⍀ : L ,1 ¬ L␦,␤.21Ž.

This means that the transmutation operator fromŽ. 4 to Ž 16 . in C␮ is

y1 ␦ ␦ T* s ⍀ T : R ª L␦,␤␦, since T*R s LT,␤*,Ž. 22 352 KIRYAKOVA AND AL-SAQABI illustrated by the following commutati¨e diagram:

␦ R 6 CC␮ ␮

T T

6 6 6 CC␮ ␮ L␦,1

⍀y1 ⍀y1

6 6 6 CC␤␮␤␮ L␦,␤

For simplicity, we consider first Eq.Ž. 16 with ␤ s 1. Denoting Ty˜Ž. x [ yxŽ.,Tf˜ Ž. x [ fx Ž., from relationŽ. 20 we observe that T transforms the simpler Eq.Ž. 4 into the E-K equation, namely,

␦ ˜ T : ˜˜yxŽ.y␭Ryx Ž.sfx Ž.¬yx Ž.y␭Lyx␦,1 Ž.sfx Ž., that is, it transforms also the known solutionŽ. 5 into the sought solution of Ž.16 ,

x yxŽ.Ty Ž. x Tfx˜˜ Ž.␭ Žx tE .␦y1 ␭ Žxtftdt .␦ Ž. s˜ s ½5qH y ␦,␦ y 0 x y␥␥␦y1␦ sfxŽ.q␭xxH ŽytE .␦,␦ ␭ Žxyttftdt . Ž.,23 Ž . 0 which is expressionŽ. 17 with ␤ s 1. The details of evaluation ofŽ. 23 can be seen inwx 20 . To transfer this result to the case of arbitrary ␤ ) 0, we apply mapping Ž.19 ,

1 ␤␤ ⍀y :yxŽ.¬yx Ž .[ˆyx Ž., fx Ž.¬fx Ž .[fxˆ Ž. and use relationŽ. 21 . Thus,

y1 y1 ⍀ yxŽ.q␭⍀ Lyx␦,1 Ž.sˆˆyx Ž.q␭Lyx␦,␤ Ž.sftˆ Ž.,

Ž. y1 Ž. Ž. Ž . that is, the image ˆyxs⍀ yxgC␤␮ of yx given by 23 , is the unique solution of FIEŽ. 16 with arbitrary ␤ ) 0:

x ␦ 1 ␦ y␤␥␤␤y ␤␤ ␤␥␤ ˆyxŽ.sfxˆˆ Ž.q␭xxH ŽytE .␦,␦ ␭ Žxyttftdt . Ž.Ž .. 0 This ends the proof. FRACTIONAL DIFFERINTEGRAL EQUATIONS 353

By means of a similar transmutation operator T we find the solutions of FDEs involving Erdelyi´ ᎐Kober derivatives, of the form

y␤␦ ␣, ␦ D s x D␤ , ␦)0, as transformations Ty˜Ž. x of the known solutionsŽ. 7 of Ž. 6 . The following auxiliary result is used.

Žn. LEMMA. In C␮ , ␮ Gy1 the following relation between factional deriä- ␦ tiës D and D, ␤ s 1 ïa the transmutation operator 0, ␣q␦ Ži. Ži. T s I1 : C␮␮¬C , ␮Gy1, i s 0,1,2,...Ž. 24 holds: n xyk ␦ TD˜˜ yŽ. x s DTy Ž. x y Ýbk .25Ž. ⌫Ž.␣␦k1 ks1qyq The proof is given inwx 20 . Then, we can establish the following

THEOREM 2. The general solution of the E-K fractional differential equa- Ä4 tion with f g C␤␮ , ␮ G max 0, y␣ y ␦ y 1, y␤␦ ␣, ␦ x D␤ yxŽ.y␭yx Ž.sfx Ž. Ž.26 Žn. Ä4 in the space C␤␮ , ␮ G max 0, y␣ y ␦ y 1, n g ގ, n y 1 - ␦ F n, has the form n ␤Ž␦yj. ␭ ␤␦ yxŽ.sÝbxj E␦,␣q2␦yjq1Ž.x js1

x ␦ 1 ␦ y␤Ž␣q␦. ␤␤y ␤Ž␣q␦. ␤␤ ␤ qxxHŽ.yttE␦,␦␭ Ž.Ž.Ž.xytftdt 0 Ž.27

with arbitrary constants bj, j s 1,...,n, depending on the initial ¨alue data. Proof. For simplicity we assume that ␤ s 1. Consider R-L fractional differential equationŽ. 6 and apply to both sides the E-K transmutation operatorŽ 24 . , denoting yx Ž.sTy˜ Ž. x , Fx Ž.sTF˜ Ž. x . According toŽ. 25 , the result is n xyk D yxŽ.y␭yx Ž.sFx Ž.qÝbk [fxŽ., ⌫Ž.␣ ␦ k 1 ks1qyq that is, Eq.Ž. 26 . Thus, its solution is the T-image of solutionŽ. 7 with n xyk y1 y1 Fx˜Ž.sTFx Ž.sTfx Ž.yÝbk . ½5⌫Ž.␣␦k1 ks1qyq 354 KIRYAKOVA AND AL-SAQABI

After complicated but routine manipulationsŽ seew 20, Proof of Theo- rem 2x. , we obtain the solution

n ␦yj ␦ yxŽ. TbxEÝj ␦,1 ␦ jŽ.␭x s ½ q y js1

x ␦y1 ␦ qHŽ.xytE␦,␦␭ Ž.Ž.xytFtdt˜ 0 5 in the formŽ. 27 with ␤ s 1. The case of arbitrary ␤ ) 0 follows by the y1 transformation ⍀ . The conditions ␦ ) 0, f g C␤␮ , ␮ ) y␣ y ␦ y 1 ensure the convergence of integralŽ. 27 .

3. SOLUTIONS TO HYPERGEOMETRIC INTEGRAL EQUATIONS OF THE SECOND KIND

In this paper we extend the method fromwx 20 to the case of Eq.Ž. 13 with both ␦ / 0, ␯ / 0 as well as to hypergeometric integral equationsŽ. 14 of the second kind. We show that solving differintegral equationsŽ. 13 , similar toŽ. 8 and involving both E-K fractional integrals and derivatives, can be easily reduced to solving integral equations in¨ol¨ing products of two differ- ent E-K integrals. According towx 10, Theorem 1.2.10 , each product of two commuting E-K fractional integralsŽ. 11 can be represented by means of a generalized Ž.2-tuple fractional integral, involving a Gauss hypergeometric function Ž for . simplicity we let here ␤12s ␤ s 1,

␥1,␦1␥2,␦2 Ž␥1,␥2.,Ž␦1,␦2. I11IyxŽ.sIŽ1,1.,2 yxŽ.

1 2,0 ␥1122q␦ ,␥ q␦ sHG2,2 ␴ yxŽ.␴ d␴ 0 ␥12,␥

␥ ␦12q␦y1 1␴2Ž.1y␴ sH 0⌫Ž.␦12q␦

=21221112FŽ.␥q␦y␥,␦;␦q␦;1y␴ yxŽ␴.d␴, Ž.28 FRACTIONAL DIFFERINTEGRAL EQUATIONS 355 called the hypergeometric fractional integral. The same can be also written as an integral from 0 to x by the substitution ␴ s trx. It is supposed that Ž. ygC␮,␥1, 2 Gy␮y1. Operators 28 have been considered by Kiryakova wx10 as special cases Ž.m s 2 of the so-called generalized Ž m-tuple . fractional integrals. For the first time, the methods of fractional differintegration in studying hypergeometric integral equations of the first kind, involving operatorsŽ. 28 , have been applied by Lovewx 11, 12 . He has found necessary and sufficient conditions for the uniqueness of the solutions and has represented them explicitly by means of operators of the same typeŽ. 28 . Hypergeometric fractional integrals have been also considered by Kalla and Saxenawx 8 , Saigowx 17 , Srivastava and Buschmanwx 22 , etc. Here we solve some hypergeometric integral equations of the second kind, that involve products of two E-K fractional integrals of the form

␦␣q␦,␦ ␯␣y␯,␯ ␦q␯ ␣q␦q␯,␦ ␣y␯,␯ H yxŽ.sŽ.xI11Ž.xI yxŽ.sx I11I yxŽ.

␦q␯Ž␣q␦q␯,␣y␯.,Ž␦,␯. sxIŽ1,1.,2 yxŽ.; ygC␮*,␮*G␯y␣y1, Ž.29 with arbitrary ␣ g R and fractional multiorder of integration ␦ ) 0, ␯ ) 0. To solve the Volterra integral equation of the second kind

yxŽ.y␭Hyx Ž.sF Ž.x, we reduce it to Eq.Ž. 4 , using the transmutation method again.

Ž. THEOREM 3. The unique solution y x g C␮, ␮ G ␯ y ␣ y 1 of the hypergeometric integral equation of the second kind

yxŽ.y␭Hyx Ž.

␦q␯y1␣ ␯ xŽ.x tty ␯␣ y syxŽ.y␭xyH 0 ⌫Ž.␦q␯ t =21Fy␯y␦,␦;␦q␯;1y ytŽ. dtsF Žx . Ž30 . ž/x 356 KIRYAKOVA AND AL-SAQABI

with F g C␮, ␮ G ␯ y ␣ y 1, is gi¨en by the formula

␯ ␭Ž.Ž1 ⌫ ␯ 1 .ϱ Ž.1 k x k ⌫ Ž␯ k ␣ . y q ␦␣ y y q yxŽ.sF Ž.xq xyyÝ ⌫Ž.␦ k!⌫Ž.␯ k 1 ks0 yq x 1 2␦q2␯yky1 =HTyFŽ.Žtxyt . 0

␭ ␦q␯ =E␦q␯,2␦q2␯yk Ž.xytdt.31Ž.

Proof. Now we observe that the E-K fractional integral

␯y␣ ␯ , ␦ T s x I1 : C␮␮, ␮ Gy1ªC *, ␮*G␯y␣y132Ž.

␦ ␯ ␦ ␯ is a transmutation operator from the R-L integral R q s R R into the hypergeometric fractional integral H, i.e.,

␦q␯ ␦q␯ T : R ª H, since TR s HT in C␮.33Ž.

Indeed, the operational techniques known for the E-K fractional integrals wx10, Chap. 1 , give in C␮, ␮ Gy1,

␦q␯ ␦ ␯ ␦ 0, ␦ ␯ 0,␯ ␦q␯ ␯, ␦ 0,␯ R s R R s Ž.Ž.x I11x I sx I 11I

and together withŽ. 29 ,

␦q␯ ␯y␣ ␯ , ␦ ␦q␯ ␯ , ␦ 0, ␯ ␯y␣ ␦q␯ 2␯q␦, ␦ ␯, ␦ 0, ␯ TR s Ž.Ž.x I111111x I I sx x I I I

␦q␯␯y␣2␯q␦,␦0, ␯ ␯, ␦ ␦q␯ ␯q␦q␣, ␦ ␯y␣ 0, ␯ ␯, ␦ sxxI111II sx I 1x I 11I

␦q␯␯q␦q␣,␦␣y␯,␯␯y␣␯,␦ sŽ.xI11IŽ.xI 1sHT.

Consider the R-L integral equation of formŽ. 4 , replacing ␦ ¬ ␦ q ␯, fx˜˜Ž.¬F Ž.x,

␦ ␯ ˜˜yxŽ.y␭Ryxq Ž.sF˜ Ž.x Ž.34

and apply transformation T, denoting yxŽ.sTy˜ Ž. x , F Ž.x sTF˜ Ž.x . Rela- tionŽ. 33 gives then

␦ ␯ yxŽ.y␭ ŽTR q .Ž.˜yxsyx Ž.y␭Hyx Ž.sF Ž.x, T i.e., Eq.Ž. 34 ¬Eq. Ž. 30 . FRACTIONAL DIFFERINTEGRAL EQUATIONS 357

Then, the sought solution yxŽ.is the image of Ž. 5 with ␦ ¬ ␦ q ␯, fx˜˜Ž.¬F Ž.x, and since

␦y1 ␯ x Ž.x ␶␶ ␯y␣␯,␦ y␦y␣ y T⌽Ž.xsxI1 ⌽Ž.xsx H ⌽Ž.␶d␶; 0 ⌫Ž.␦

1 y1 ␯,␦␯y y␣␯y1 ,␦␣y␯ TFŽ.xsŽ.Ix11 Ž .F Ž.xsDx FŽ.x

y␯␦␦q␣ sxDxxFŽ.x,35Ž. we have

x yxŽ.TF˜˜ Ž.x ␭ Žx tE .␦q␯y1 ␭Žxt .␦q␯ F Ž.tdt s ½5qH y ␦q␯,␦q␯ y 0 x ␦␣1 sFŽ.xq␭xTyyH yFŽ.tKx Ž,tdt . , 0 with an inner integralŽ. after interchanging both integrations

␦y1 x Ž.x y ␶ ␶␶␯ ␦q␯y1 ␭␶ ␦q␯␶ KxŽ.,tsH Ž.ytE␦q␯,␦q␯Ž.ytd t ⌫Ž.␦

y to be evaluated. By the substitutions x y t [ y, ␶ y t [ ␪ <0 , d␶ s d␪ we put KxŽ.,t in the form

␦y1 y Ž.y y ␪ ␯ ␪␪␦q␯y1 ␭␪ ␦q␯␪ KsH xyŽ.yy E␦q␯,␦q␯wxd . 0 ⌫Ž.␦

The evaluation of the above integral can be done by replacing the term ␯ w xyŽ.yy␪xby means of its series expansionŽ. Newton binomial series , interchanging the summation and the fractional integration and application of the known formula for fractional integrals of M-L functions Žseew 2, Eq. 6xw , or 10, Eq.Ž. E.28x. :

␮ ␤y1 ␭ ␣ ␤q␮y1 ␭ ␣ ␮ R Ä4y E␣,␤␣Ž.y s y E ,␤q␮Ž.y , ) 0.Ž. 36 Thus,

␯ k k Ž.Žy1 ⌫ ␯ q 1 .ϱ Ž.y1 x ⌫ Ž␯ y k q ␦ . KxŽ.,ts Ý ⌫Ž.␦ k!⌫Ž.␯k1 ks0 yq 2␦q2␯yky1 ␭␦q␯ =Ž.xytE␦q␯,2␦q2␯yk Ž.xyt and routine calculations lead further to resultŽ. 31 . 358 KIRYAKOVA AND AL-SAQABI

Let us note that specific choices of the right-hand side FŽ.x and parameters ofŽ.Ž. 29 , 30 , turn Ž. 31 into simpler and efficient expressions Ž.see Example 3 . Theorem 3 is not only a self-contained result, but it happens to be useful also in solving differintegral equations of the formŽ. 13 .

4. FRACTIONAL DIFFERINTEGRAL EQUATIONS INVOLVING BOTH E-K INTEGRALS AND DERIVATIVES Ž. Žn. THEOREM 4. The solutions y x g C␮ , ␮ Gy␦y␣y1, n g N, n y 1 - ␦ F n of the fractional differintegral equation

y␦ ␣ , ␦ ␯␣y␯,␯ x D11yxŽ.y␭xI yxŽ.sfx Ž., real ␣ , ␦ ) 0, ␯ ) 037Ž.

with f g C␮, ha¨e the form n xy␣yj ␣ , ␦␦ yxŽ.sÝbj qIxfx1 Ž. ⌫Ž.␦ j 1 js1yq ␯ Ž.Ž1⌫␯1 .ϱ Ž.1kxk⌫ Ž␯k␦ . yq␦␣ y yq q␭ xyy=Ý и ⌫Ž.␦ k! ⌫Ž.␯k1 ks0 yq ␣q␦y␯ ଙ2␦q2␯yky1 ␭␦q␯ =Ä4xfxŽ. xE␦q␯,2␦q2␯ykŽ.x, Ž.38 where x F ଙ GxŽ.sHFx ŽytGt . Ž. dt Ž.39 0

denotes the Duhamel con¨olution and bj, j s 1,...,n, are arbitrary constants. Proof. To reduce differintegral equationŽ. 37 to an integral one, namely to Eq.Ž. 30 , we use the relations between the E-K integrals and derivatives Ž.Ž. ␣,␦␣,␦Ž. Ž. 11 , 12 : DIyx11 syx and n xy␣yj ␣,␦␣,␦ IDyx11Ž.syx Ž.yÝbj , ⌫Ž.␦j1 js1yq

Žn. ny1-␦Fn,ygC␮,␮Gy␣y1.Ž. 40

␣,␦␦ Then, applying the E-K fractional integral Ix1 to both sides ofŽ. 37 , we get

␣ , ␦ ␦ y␦ ␣ , ␦ ␣,␦ ␦ ␯␣y␯,␯ ␣,␦ ␦ Ž.Ž.I11x x D yxŽ.y␭ Ž.I 11x Ž.xI yxŽ.sI 1xfx Ž., FRACTIONAL DIFFERINTEGRAL EQUATIONS 359

and further, relationŽ. 40 and the operational properties of E-K integrals lead to

n xy␣yj ␦q␯␣q␦q␯,␦␣y␯,␯ yxŽ.yÝbj y␭Ž.xI11 I yxŽ. ž/j1⌫Ž.␦yjq1 s

␣,␦␦ sI1 xfxŽ.. Denoting n xy␣yj ␣ , ␦␦ FŽ.x[Ýbj qIxfx1 Ž.gC␮, ␮Gy␦y␣y1, ⌫Ž.␦ j 1 js1yq Ž.41

we obtain Eq.Ž 30 . , yx Ž.y␭Hyx Ž.sF Ž.x, whose solution is given byŽ. 31 . It remains to replace FŽ.x by Ž 41 . and to evaluate Ty1FŽ.t , according to Ž.35 :

n ty␣yj y1 y␯␦␦q␣ y␯␦␦q␣␣,␦␦ T FŽ.tstDttjÝ b qtDttÄ4 I1 t ½5⌫Ž.␦yjq1 js1

[T12qT. After simple calculations, since 1r⌫Ž.1 y j s 0, j s 1,...,n, we find n tDty␯␦␦Ä4yj n ty␯yj T1sÝÝbjjsиии s b s 0 ⌫Ž.␦j1⌫ Ž.1j js1yqjs1y and

y␯ ␦ ␦q␣ y␣y␦ ␦ ␣ ␦ y␯ ␣q␦ ␣q␦y␯ T2s t Dtt Žt R t .t ft Ž.st t ft Ž.st ft Ž., ␦ ␦ since DR sId. In this way we obtain the solutions ofŽ. 37 in the form n xy␣yj ␣ , ␦␦ yxŽ.sÝbj qIxfx1 Ž. ⌫Ž.␦ j 1 js1yq ␯ ␭Ž.Ž1⌫␯1 .ϱ Ž.1kxk⌫ Ž␯k␦ . yq␦␣ y yq q xyyÝ ⌫Ž.␦ k!⌫Ž.␯k1 ks0 yq x ␣ ␦ ␯ 2␦q2␯yky1 =½Htftxqy Ž.Žyt . 0

=E ␭Ž.x tdt␦q␯ , ␦q␯,2␦q2␯yk y 5 360 KIRYAKOVA AND AL-SAQABI equivalent toŽ. 37 , where the notation ଙ for the Duhamel convolution Ž. 39 is used. The arbitrary constants bj, j s 1,...,n, depend on the initial Ž␦ j. value conditions y y Ž.0,js1,...,n, ny1-␦Fn. Again by the transformation ⍀y1, fromŽ. 38 we can obtain the solutions of FDIE of formŽ. 37 but involving E-K integrals and derivatives with arbitrary parameter ␤ ) 0.

5. EXAMPLES

To demonstrate the efficiency of the solutions from Theorems 1᎐4, we give some examples. First we take particular right-hand sides fŽ. x .

EXAMPLE 1. SolutionŽ. 17 of the E-K integral equation Ž. 16 of the second kind takes the form

␤p Ž.i for fx Ž .sx ,p)y␥y1,

␤ p ␭⌫ ␥ ␤␦ ␭␤␦ yxŽ.sx 1q Ž.pq q1xE␦,␦qpq␥q1Ž.Ž.x ;42

Ž. Ž. Ž ␤. ii for fxsE␮, ␯ ␣x , arbitrary ␮, ␯, ␣ g R,

ϱ k! k ␤␤Ž␦y␥. ␤␤␦ yxŽ.Ž.sE␮,␯␦␣x q␭x Ý Ž.␣xE,␦k1 Ž␭x .; ⌫Ž.␯k␮qq ks0q Ž.43

Ž.iii if in the above case ␮ s ␯ s 1, we obtain the solution for ␤ fxŽ.sexp Ž␣ x .,

ϱ k ␣ ␤␤␭ Ž␦y␥.␣ ␤␤␭␦ yxŽ.Ž.sexp x q x ÝŽ.xE␦,␦qkq1 Žx .Ž..44 ks0

␤p EXAMPLE 2. Fractional differential equationŽ. 26 with fx Ž.sx ,p) y␣y␦y1 has solutionsŽ. 27 of the form

n ␤Ž␦yj. ␭ ␤␦ yxŽ.sÝbxj E␦,␣q2␦yjq1Ž.x js1 ⌫␣ ␦ ␤Ž␦qp. ␭␤␦ qŽ.qqpq1x E␦,␣q2␦qpq1 Ž.Žx .45. FRACTIONAL DIFFERINTEGRAL EQUATIONS 361

p EXAMPLE 3. Hypergeometric equationŽ. 30 with F Ž.x s x has a solu- tionŽ. 31 of the explicit form

⌫Ž.Ž␯ 1 ⌫ ␦ ␣ p 1 .Ž⌫ ␣ ␯ p 1 . p ␦p ␯ q q q q y q q yxŽ.sxq␭xqŽ.y1 ⌫Ž.Ž␦⌫␣qpq1 . ϱ Ž.Ž1k⌫␯ k ␣ . y yq ␦q␯ =Ý E␦␯,2␦ ␣ k p 1Ž.Ž.␭x .46 k!⌫Ž.␯ k 1 qq y q q ks0 y q

It is interesting to consider also special cases of Erdelyi´ ᎐Kober operators in Eqs.Ž.Ž. 13 , 14 . Naturally, if ␦ s ␤ s 1, ␥ s 0, ␣ sy1, Eqs.Ž.Ž. 16 , 26 turn into the simplest onesŽ.Ž. 4 , 6 used as a base here.

EXAMPLE 4. Consider the so-called Dzrbashjan᎐Gelfond᎐Leontie¨ ŽD- G-L.Ž.Ž.integrals and deri¨ati¨es, special cases of the E-K operators 11 , 12 , studied by Dimovski and Kiryakovawx 1 and Kiryakova w 10, Chap. 2 x :

␯y1,1r␳ ␳Ž1r␳. ␯y1,1r␳ l␳,␯␳[ xI sx I ,

␯y1,1r␳ y1 y␳Ž1r␳. ␯y1r␳y1,1r␳ d␳,␯␳[Dx sx D␳ .47Ž.

Ž. Ýϱ k For analytic functions yxs ks0 axk these operators have also series representations:

ϱ ⌫Ž.␯ k ␳ q r kq1 lyx␳,␯ Ž.sÝaxk , ⌫Ž.␯ Ž.k 1 ␳ ks0q q r ϱ⌫Ž.␯k␳ qrky1 dyx␳,␯ Ž.sÝaxk .48Ž. ⌫Ž.␯ Ž.k 1 ␳ ks1q y r

The corresponding D-G-L integral and differential equations

yxŽ.y␭lyx␳,␯␳ Ž.sfx Ž., dyx,␯ Ž.y␭yx Ž.sfx Ž. are special cases ofŽ.Ž. 16 , 26 with ␦ s 1r␳, ␤ s ␳, ␥ s ␯ y 1, ␣ s ␯ y 1r␳ y 1. Then Theorems 1, 2 give their solutions

x ␭y␳Ž␯y1.␳␳1r␳y11␭␳␳r␳ yxŽ.Ž.sfxq xxH Ž.ytE1r␳,1r␳ Ž.xyt 0 =t␳Ž␯y1.ftŽ. dt Ž␳ . Ž.49 362 KIRYAKOVA AND AL-SAQABI and respectively,

n 1y␳ j ␭ yxŽ.sÝbxj E1r␳,␯q1r␳yjŽ.x js1

x y␳Ž␯y1.␳␳1r␳y11 ␳Ž␯y1. ␭␳␳r␳␳ qxxHŽ.yttE1r␳,1r␳ Ž.Ž.Ž.xytftdt. 0 Ž.50

EXAMPLE 5. The so-called Rusheweyh derivatives, defined by means of the Hadamard productŽ.Ž convolution seewx 10, Chap. 5 . x 1 D␣ yxŽ. yxŽ. Dyxy1, ␣ Ž.,␣)0 s 1q␣ ( s 1 ½5Ž.1yx ⌫Ž.␣q1 Ž.51 are often used in analytic functions theory. The corresponding fractional ‘‘differential’’ equations

ϱϱ y␣␣ ki xDyxŽ.y␭yx Ž.sfx Ž.for yx Ž.sÝÝaxki,fxŽ.s cx ks0is0 Ž.52 have the form

ϱϱ⌫Ž.␣k1xy␣ qqki ÝÝakiy␭xscx ⌫Ž.␣1k! ks0 qis0 and according to Theorem 2, their solution are

n ϱ ␣yj ␭ ␣␣⌫ ␣ qi␭␣ yxŽ.sÝÝbxj E␣,2␣yjiŽ.x q c Žqix . E␣,2␣qi Ž.x . js1 is0 Ž.53

From the above, interesting relations between coefficients aki, c , k, i s 0, 1, 2, . . . , follow. Especially, for ␣ s 1 we obtain the following solution of the first order y1 y1, 1 Ž. Ž. Ž. Ž. Ž. differential equation xD1 yxy␭yx syЈx y␭yx sfx, Ž. Ýϱ i fx[ is0cxi : ϱ ␭ iq1 ␭ yxŽ.Ž.Ž.sy0 exp x q xciÝi!xE1, iq2Ž.x.54 Ž. is0 FRACTIONAL DIFFERINTEGRAL EQUATIONS 363

It follows easily also from the well known solution yxŽ.sbexp Ž␭ x .q expŽ␭ x .ଙ fx Ž., if we replace fx Ž.by its series and evaluate the integrals under the summation sign as Mittag᎐Leffler functionsŽ to this end formula Ž. Ž. Ž.. 36 is used again with exp ␭ x s E1, 1 ␭ x . EXAMPLE 6. Consider differintegral equationŽ. 37 with arbitrary ␣ g R and ␦ s ␯ s 1 « n s 1, i.e., d x Ž ␣ 1. ␣ 11␣␣1 xxyxy q q Ž.yxtytdtyHyŽ.sfx Ž.. dx 0 FormulaŽ. 38 gives for the solutions Žb s y Ž..0

␣ 1 1 ␣ 1 yxŽ.sxyy ½bqRŽ.xqfxŽ.

ϱŽ.xk y␣ 3yk 2 y␭ÝÄ4xfxŽ.ଙ xE2,4 kŽ.␭x k! y 5 ks0 Ž. y␣ and especially, if fxsx gCy␣y1, then i 1 ϱϱ Ž.Ž.1k ␭x2 ␣1 ␣13␣ y yxŽ.sy Ž.0xyyq xyqy␭xyÝÝ 2 k!⌫Ž.2i 5 k ks0is0 q y

gCy␣y1.55Ž. The expressions in the sample formulasŽ.Ž.Ž.Ž. 42 ᎐ 46 , 53 ᎐ 55 allow efficient numerical procedures for their evaluation.

ACKNOWLEDGMENTS

This paper has been partially supported by Research Project SM-115r1995Ž Research Administration of Kuwait University.Ž and Research Project 606r1996 NSF, Bulgarian Ministry of Education, Science and Technologies. .

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