Solving fractional Fredholm integro–differential equations using Legendre wavelets

Abstract

This paper presents a Legendre wavelet spectral method for solving a type of fractional Fredholm integro–differential equations. The fractional derivative is defined in the Caputo–Prabhakar sense. The derivative of Prabhakar consists of an integro–differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes the Riemann–Liouville and Caputo fractional operators. Moreover, it has many applications in several fields of computational physics. We first derive a matrix method to solve linear problems. In this method, the given linear problem is reduced to a linear system of algebraic equations. The detailed convergence analysis of the proposed matrix method is given. An iterative matrix method is then constructed for nonlinear problems. The nonlinear problem is first replaced with a sequence of linear problems by utilizing the quasilinearization technique. Then, this sequence of problems is successively solved using the matrix method. Numerical examples are included to demonstrate the efficiency and accuracy of the proposed methods.