A new recursive spectral Tau method on system of generalized Abel-Volterra integral equations

TL;DR

This paper introduces a recursive spectral Tau method using fractional canonical polynomials based on M"untz-Legendre polynomials to efficiently solve systems of generalized Abel-Volterra integral equations with high accuracy.

Contribution

It develops a novel recursive spectral Tau approach with fractional canonical polynomials tailored for singular integral equations, improving computational efficiency and accuracy.

Findings

Achieves spectral accuracy in $L^{inity}$ norm.

Uses low-dimensional algebraic systems for parameter calculation.

Demonstrates effectiveness through numerical examples.

Abstract

This paper provides an efficient recursive approach of the spectral Tau method to approximate the solution of system of generalized Abel-Volterra integral equations. In this regards, we first investigate the existence, uniqueness as well as smoothness of the solutions under assumption on the given data. Next, from a numerical perspective, we express approximated solution as a linear combination of suitable canonical polynomials which are constructed by an easy to use recursive formula. Mostly, the unknown parameters are calculated by solving a low dimensional algebraic systems independent of degree of approximation which prevent from high computational costs. Obviously, due to singular behavior of the exact solutions, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regards, we develop a new fractional order canonical polynomials using M\"untz-Legendre polynomials which have a same asymptotic behavior with the solution of underlying problem. The convergence analysis is discussed, and the familiar spectral accuracy is achieved in $L^{\infty}$ norm. Finally, the reliability of the method is evaluated using various problems.