Generalised Müntz Spectral Galerkin Methods for Singularly Perturbed Fractional Differential Equations

Abstract

A family of orthogonal generalised Müntz-Jacobi functions is introduced and used for solving singularly perturbed fractional differential equations. Such basis functions can provide much better approximation for boundary layers or endpoint singularities than usual polynomial bases. The fractional integrals and derivatives of generalised Müntz-Jacobi functions are accurately calculated. The corresponding Petrov-Galerkin and Galerkin methods are very efficient. Numerical examples demonstrate a significant improvement in the accuracy of the methods.