Numerical Solution of Fractional Order Fredholm Integro-differential Equations by Spectral Method with Fractional Basis Functions

This paper introduces a new numerical technique based on the implicit spectral collocation method and the fractional Chelyshkov basis functions for solving the fractional Fredholm integro-differential equations. The framework of the proposed method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with non-smooth solutions.