Some triple integral equations

Abstract

φ(s>> *} = 0, 0 Φ(s); x] = Ux a η, δ > 0, 0, are real parameters, /2O&) is a known function, Φ(s) is to be determined and (3) m{h(x); s} = H(s), Tl^His); x) = h(x) , denote the Mellin transform of h(x) and its inversion formula respectively. The above equations are an extension of the dual integral equations solved in a recent paper by Erdelyi [2] by means of a systematic application of the Erdelyi-Kober operators of fractional integration [4]. Using the properties of some slightly extended forms of the Erdelyi-Kober operators we show, in a purely formal manner, that the solution of the triple integral equations can be expressed in terms of the solution of a Fredholm integral equation of the second kind. Srivastav and Parihar [5] have solved a very special case of the equations by a completely different method from that used in this paper. The method of solution employed here will be seen to follow closely that used by Cooke [1] to obtain the solution to some triple integral equations involving Bessel functions; indeed Cooke's equations may be regarded as a special case of equations (1) and (2) and it is shown that a solution of his equations can be readily obtained from that presented in this paper.