An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions

TL;DR

This paper introduces a spectral Tau method using fractional canonical basis polynomials to efficiently solve third-kind Volterra integral equations with non-smooth solutions, maintaining high accuracy.

Contribution

A novel fractional basis polynomial set (FC-polynomials) is developed for spectral methods, enabling accurate solutions for non-smooth integral equations.

Findings

Method achieves high accuracy with non-smooth solutions.

Numerical results outperform existing methods.

Convergence analysis confirms method's reliability.

Abstract

This paper is concerned with the numerical solution of the third kind Volterra integral equations with non-smooth solutions based on the recursive approach of the spectral Tau method. To this end, a new set of the fractional version of canonical basis polynomials (called FC-polynomials) is introduced. The approximate polynomial solution (called Tau-solution) is expressed in terms of FC-polynomials. The fractional structure of Tau-solution allows recovering the standard degree of accuracy of spectral methods even in the case of non-smooth solutions. The convergence analysis of the method is studied. The obtained numerical results show the accuracy and efficiency of the method compared to other existing methods.