On Solving a Singular Volterra Integral Equation

Abstract

In the paper, a general solution of the singular Volterra integral equation of the second kind is found. The feature of the considered integral equation lies in the fact that the integral of its kernel does not tend to zero as the upper limit approaches the lower one, thus making the Picard method inapplicable. It is shown that the corresponding homogeneous integral equation has a non-zero solution, which is found in an explicit form. Such integral equations arise, for example, in boundary value problems of heat conduction in degenerate regions, where the boundaries change with time \cite{article1}-\cite{article5}. Additionally, these equations appear in mathematical modeling of thermophysical processes in the electric arc of high-current switching devices.