Runge-Heun-Kutta Methods for Nonlinear Fractional Order Differential Equations

Abstract

In this paper, we develop Runge-Heun-Kutta methods for the numerical solution for fractional order differential equations (FDE), whereby both explicit and implicit methods are developed. Based on Heun’s insight of interpreting Runge’s method as a generalization of Gaussian quadrature, we develop methods that are backwards compatible with classic Runge-Kutta methods such as Heun’s method with third order truncation error. Finally, we devise a new approach for treating any nonlinear FDE with multiple fractional order derivatives of any arbitrary order. The methods are tested with some of the more difficult benchmark problems for nonlinear FDE.