Decomposition of an Integral Operator by Use of Mikusiński Calculus

Abstract

Recently T. R. Prabhakar used fractional integrals in order to obtain explicit solutions to a convolution integral equation in which the kernel involved a confluent hypergeometric function. Decomposition of the integral operator into fractional integrals and exponential functions plays a role in the development and, following the ideas of A. Erd61yi, this decomposition is treated here in a clearer format from the standpoint of Mikusifiski operators. Further, the conditions for existence and uniqueness of the solution are conveniently displayed.In a recent paper by T. R. Prabhakar [4] fractional integrals are used in order to obtain explicit solutions of a convolution integral equation of the formin which the kernel is of the form K(a; b; 20 [tb-/F(b)]lF(a; b; 2t) and 1Fa denotes the confluent hypergeometric function with Re b > 0. Decomposition of the operator Ka, into a product of simpler operators plays a role in the development. Following the ideas of A. Erd61yi [2], this equation can be studied from the standpoint of Mikusifiski calculus and the decompositions can be presented in a clearer format.The following correspondences to Mikusifiski operators are needed" K(a;b;2t)s"-b(s-2) -, Reb>0, Re2>0;It"-et/F(#)] (s 2)-", Re # > 0.In view of the development by Erd61yi [2], these can be obtained from tables of Laplace transforms [3]. Since by Kummer's transformation, 1Fa(a;b; -20=e-Xt F(b-a'b'20 we see that the first relation also holds for Re 2 < 0. For 2 0 in the second correspondence we obtain l" s -u, where I" denotes the Riemann-Liouville fractional integral operator of order / for Re/ > 0. This can be extended to Re/ =< 0 in the usual manner by choosing a positive integer n such that Re(n +/0 > 0 and letting I" I-"I "+" in which I-" denotes a differentiation operator such that the property I-"f s"fis retained.We note that -(d) f (u) du -(s 2)-"f.

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