Fractional Order Runge-Kutta Methods

TL;DR

This paper introduces a new class of fractional order Runge-Kutta methods for solving fractional differential equations, extending classical schemes with modifications to handle fractional derivatives and demonstrating their effectiveness through analysis and experiments.

Contribution

It develops explicit and implicit fractional Runge-Kutta methods based on Caputo derivatives, including convergence analysis and numerical validation.

Findings

Methods are effective for fractional differential equations.

Numerical experiments confirm robustness and accuracy.

Convergence of the proposed schemes is established.

Abstract

This paper investigates, a new class of fractional order Runge-Kutta (FORK) methods for numerical approximation to the solution of fractional differential equations (FDEs). By using the Caputo generalizedTaylor formula and the total differential for Caputo fractional derivative, we construct explicit and implicit FORK methods, as the well-known Runge-Kutta schemes for ordinary differential equations. In the proposed method, due to the dependence of fractional derivatives to a fixed base point $t_0,$ we had to modify the right-hand side of the given equation in all steps of the FORK methods. Some coefficients for explicit and implicit FORK schemes are presented. The convergence analysis of the proposed method is also discussed. Numerical experiments clarify the effectiveness and robustness of the method.