Systems of Integral Equations with a Degenerate Kernel and an Algorithm for Solving Them Using the Maple Math Software

Abstract

The mathematical literature contains a good description of a scalar integral equation with a degenerate kernel (1), in which all of the written functions are scalar quantities. We are not aware of publications in which systems of integral equations of type (1) with kernels in the form of a product of matrices would be considered in detail. It is usually said that the technique for solving such systems is easily transferred from a scalar case to a vector one. For example, in A.L. Kalashnikov’s text book “Methods for Approximate Solution of Integral Equations of the Second Kind,” systems of equations with degenerate kernels are briefly described, in which the products of scalar rather than matrix functions play the role of degenerate kernels. However, as the simplest examples show, the generalization of the ideas of the scalar case for the case of integral systems with kernels represented by the sum of products of matrix functions is rather unclear, although in this case the idea of reducing an integral equation to an algebraic system is also used. In our opinion, the process of obtaining the conditions for the solvability of an integral system in the form of orthogonality conditions, based on the conditions for the solvability of the corresponding algebraic system, has not been described previously. Bearing in mind the great capabilities of the theory of integral equations in applied problems, we considered it necessary to give a detailed scheme for solving integral systems with degenerate kernels in a multidimensional case and to implement this scheme in the Maple math software. It should be noted that only scalar integral equations are solved in Maple using the intsolve procedure. Since we could not find a similar procedure for solving systems of integral equations, the have developed our own procedure.