Solution to the Volterra integral equations of the first kind with piecewise continuous kernels in class of Sobolev-Schwartz distributions

TL;DR

This paper establishes conditions for the existence and uniqueness of solutions to Volterra integral equations of the first kind with piecewise continuous kernels within Sobolev-Schwartz distribution theory, and proposes an approximation method.

Contribution

It introduces new sufficient conditions for solutions' existence and uniqueness and develops an asymptotic approximation and refinement method for generalized solutions.

Findings

Derived sufficient conditions for solution existence and uniqueness.

Constructed an asymptotic approximation of generalized solutions.

Proposed a successive approximation method for solution refinement.

Abstract

Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.