Solution to the Volterra Matrix Equation of the 1st kind with Piecewise Continuous Kernels

TL;DR

This paper develops an algorithm for solving Volterra matrix equations of the first kind with piecewise continuous kernels, proving existence, uniqueness, and asymptotic behavior of solutions relevant to dynamic systems.

Contribution

It introduces a new method for constructing asymptotics and establishes conditions for the existence and uniqueness of solutions with jump discontinuities in kernels.

Findings

Algorithm for logarithmic power asymptotics proposed

Existence of parametric solution families proved

Sufficient conditions for solution existence and uniqueness derived

Abstract

In this text matrix Volterra integral equation of the first kind is addressed. It is assumed that kernels of the equation have jump discontinuities on non-intersecting curves. Such equations appear in the theory of evolving dynamic systems. Differentiation of such equations with jump discontinue kernels yields the new class of the Volterra integral equations with functionally perturbed argument. The algorithm for construction of the logarithmic power asymptotics of the desired continuous solutions is proposed. The theorem of existance of the parametric families of solutions is proved. Finally the sufficient conditions for existence and uniqueness of continuous solution are derived.