Solution of a Cauchy type problem for an integral equation of Volterra type with singular kernels, when the roots of the characteristic equations are complex conjugate

Abstract

In this paper, we study a two-dimensional Volterra type integral equation with a singularity and a logarithmic singularity for one variable and a strong singularity for another variable. The solution of an integral equation with special kernels in the case when the coefficients of the equation are interconnected is reduced to solving one-dimensional Volterra-type integral equations with special kernels. Using the connection of the considered integral equations with ordinary differential equations with singular coefficients, depending on the sign of the coefficients of the equation, explicit solutions of the studied two-dimensional integral equation are obtained. Depending on the roots of the characteristic equations and the sign of the equation parameters, explicit solutions of this integral equation are obtained. If the characteristic equation has complex conjugate roots, then the integral equation under study with special kernels has a single solution or explicit solutions contain two or four arbitrary functions. In the latter cases, the correct formulation has been clarified and explicit solutions of Cauchy-type problems have been obtained.