A New Spectral Mittag-Leffler Weighted Method to Solve Variable Order Fractional Differential Equations

Abstract

Background: In many studies, spectral methods based on one of the orthogonal polynomials and weighted residual methods (WRMs) have been used to convert the variable-order fractional differential equation (VO-FDEs) into a system of linear or nonlinear algebraic equations, and then solve this system to obtain the approximate solution. Objective: In this paper, a numerical method is presented for solving VO-FDEs. The proposed method is based on Chelyshkov polynomials (CPs). Methods: WRMs are used to obtain approximate solutions of the governing differential equations. In addition, a new weight function based on Mittag-Leffler functions is proposed. The proposed method is applied to a group of linear and non-linear examples with initial and boundary conditions. Results: Acceptable results are obtained in most of the examples. In addition, the effect of polynomials (such as Chebyshev, Jacobi, Legendre, Gegenbauer, Hermite, Taylor, Mittag-Leffler, and Bernstein polynomials) on the accuracy of the approximate solution is studied, and their effect was found to be minimal in most tests. The effect of the proposed weight function is also studied in comparison with the weight functions presented in WRMs, and it was found that it has a strong effect in most examples. Conclusions: The results obtained indicate that the proposed method is effective for solving equations of variable order.

Keywords

• Variable order derivative
• Weighted residual method
• Spectral method
• Mittag-Leffler weight function
• Chelyshkov polynomials