Efficient Mittag-Leffler Collocation Method for Solving Linear and Nonlinear Fractional Differential Equations

Abstract

In this paper, a new approximation method for fractional differential equations based on Mittag-Leffler function is developed. Finite Mittag-Leffler function and its fractional-order derivatives are investigated. An efficient technique for solving linear and nonlinear fractional order differential equations is developed. The proposed method combines Mittag-Leffler collocation method and optimization technique. Error estimation of the approximation is stated and proved. We present numerical results and comparisons of previous treatments to demonstrate the efficiency and applicability of the proposed method. Making use of small number of unknowns, the resulting solution converges to the exact one in the linear case and it has a very small error in the nonlinear case.