A Method for Numerical Solution of Third-Kind Volterra Integral Equations Using Krall-Laguerre Polynomials

TL;DR

This paper introduces a numerical method using Krall-Laguerre polynomials to solve third-kind Volterra integral equations, providing an alternative to analytical solutions and including an error analysis.

Contribution

It presents the first numerical approach for third-kind Volterra integral equations using Krall-Laguerre polynomials, expanding solution techniques beyond existing analytical methods.

Findings

The method effectively approximates solutions to third-kind Volterra equations.

Error analysis confirms the method's accuracy and convergence.

Numerical experiments demonstrate the method's practical applicability.

Abstract

The present study proposed a method for numerical solution of linear Volterra integral equations (VIEs) of the third kind, before only analytical solution methods had been discussed with reference to previous research and review of the related literature. Given that such analytical solutions are not almost always feasible, it is required to provide a numerical method for solving the mentioned equations. Accordingly, Krall-Laguerre polynomials were utilized for numerical solution of these equations. The main purpose of this method is to approximate the unknown functions through Krall-Laguerre polynomials. Moreover, an error analysis is performed on the proposed method.