Numerical Solution of Fractional Integro-Differential Equations With Non-Local Conditions Using Reproducing Kernel Method Based on Chebyshev Polynomials

Abstract

In this study, an improved version of the reproducing kernel method is presented, combined with shifted Chebyshev polynomials, to obtain numerical solutions of fractional order integro-differential equations with non-local conditions. One of the important steps of this method is the selection of a linear operator, and the selected linear operator consists of the fractional order derivative term of the considered problem. Another important step of this method is to use extra basis functions for non-homogeneous initial or boundary conditions of the considered problem and to define as many basis functions as the number of conditions of the problem. Although the construction of the approximate solution for both linear and non-linear cases of the considered problem is given separately, all numerical results are obtained by the iteration process. To show the robustness and efficiency of the method, the numerical results of six different problems considered in the literature are compared with alternative techniques, and also these results are presented by tables and graphs.