A fractional version of the recursive Tau method for solving a general class of Abel-Volterra integral equations systems

Abstract

This paper provides an efficient recursive approach of the spectral Tau method, to approximate the solution of a system of generalized Abel-Volterra integral equations. In this regard, we first investigate the existence, uniqueness as well as smoothness of the solutions under various assumptions 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 low-dimensional algebraic systems independent of the degree of approximation which prevents high computational costs. Obviously, due to the singular behavior of the exact solution, using classical polynomials to construct canonical polynomials, leads to low accuracy results. In this regard, we develop new fractional-order canonical polynomials using Müntz-Legendre polynomials which have the same asymptotic behavior with the solution of the 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 examples and applying it in solving a class of fractional differential equations systems.