Numerical Solution of Distributed Order Fractional Differential Equations Using Spectral Mittag-Leffler Weight Function Based on Chelyshkov Polynomials

In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the distributed-order fractional differential equation (DOFDEs) into a system of linear or nonlinear algebraic equations, and then solve this system to obtain the approximate solution. In this paper, a numerical method is presented for solving DOFDEs. The approximate solution is imposed as an orthogonal Chelyshkov polynomial with unknown coefficients. The required coefficients are obtained using WRMs, which transform the DOFDEs into a system of algebraic coefficients. The Mittag-Leffler function is proposed as a suitable weight function. The method has been applied to several numerical examples, such as oscillatory mathematical model, and the distributed order fractional Bagley-Torvik equation. Acceptable results were obtained in most tests. The proposed Mittag-Leffler weight method is compared with the WRMs such as Galerkin method, Petrov-Galerkin method and least square method, and the proposed weighted function showed more accurate results than the previous methods in most tests. The study showed that the effect of the test polynomials such as Chebyshev, Jacobi, Legendre, Gegenbauer, Hermite, Taylor, Mittag-Leffler, and Bernstein polynomials has a small impact on most tests. In addition, the impact of the distributed order on the accuracy of the solution was studied, and the results show that the distributed order has a strong impact on the accuracy of the solution, as its impact is direct on the non-homogeneous part, which leads to more complex equations than in cases where the orders are fixed.