A class of convolution integral equations
In recent years many authors have made significant contributions to the theory of convolution integral equations whose kernels involve certain special functions of one and more variables (see, for example, the first author and \textit{R. G. Buschman} [Indagationes Math. 36, 211-216 (1974; Zbl 0275.45014)], the first author [Appl. Analysis 5, 251-256 (1976; Zbl 0226.45006)], and \textit{R. G. Buschman}, the second author and \textit{K. C. Gupta} [Glasn. Mat., III. Ser. 12(32), 61-66 (1977; Zbl 0362.45003)]). Indeed, much of the earlier work on the subject of convolution integral equations with special function kernels has been systematically presented in the book by the first author and \textit{R. G. Buschman} [Convolution integral equations with special function kernels (1977; Zbl 0346.45010)]. The object of the present paper is to develop extensions of these results to hold for the convolution integral equation \[ (*)\quad\int^{x}_{0}(x-t)^{\rho -1}e^{\alpha(x-t)}H\left( \begin{matrix} \omega_ 1(x-t)\\ \omega_ r(x-t)\end{matrix} \right)f(t)dt=g(x),\quad 0\leq x<\infty,\quad Re(\rho)>0, \] where the function g(x) is prescribed such that \(g^{(\ell)}(0)=0, \ell =0,1,2,...,n-1\), being a positive integer, f(t) is an unknown function to be determined, and the kernel involves a special class of the (Srivastava-Panda) H-function of r variables [see, for example, the first author, \textit{K. C. Gupta} and \textit{S. P. Goyal}, The H-functions of one and two-variables with applications (1982; Zbl 0506.33007); p. 251, Equation (C.1) et seq.]. By assigning suitable values to the parameters occurring in the kernel of (*), the main theorem of this paper is applied to derive solutions of certain classes of convolution integral equations whose kernels involve various special functions of one and more variables. Since each of the E, F, G and H functions occurring in the kernels of the integral equations considered in this paper can be reduced to a wide variety of simpler special functions of mathematical physics and applied mathematics (and to various products and other combinations of such useful functions), the results of this paper can indeed find many more applications than those listed by the authors.
scientific article; zbMATH DE number 3875957