New fractional Lanczos vector polynomials and their application to system of Abel–Volterra integral equations and fractional differential equations

Abstract

In this paper, the recursive approach of the τ-method is developed to construct new fractional order canonical polynomials for solving systems of Abel–Volterra integral equations. Due to the singular behavior of solutions of these equations, the existing spectral approaches suffer from low accuracy. To overcome this drawback we use Müntz polynomials as basis functions which give remarkable approximation to functions with singular behavior at origin and state Tau approximation to the exact solution based on a new sequence of basis vector canonical polynomials that is generated by a simple recursive formula in terms of fractional order Müntz polynomials. The efficiency and simplicity of the proposed method are illustrated by some examples. Convergence analysis of the method is also discussed. The paper is closed by providing application of this method to a linear multi-term fractional differential equations.