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 generalized Taylor 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.

翻译：本文研究了一类新的分数阶龙格-库塔（FORK）方法，用于分数阶微分方程（FDEs）解的数值逼近。通过使用Caputo广义泰勒公式和Caputo分数阶导数的全微分，我们构造了显式和隐式FORK方法，类似于常微分方程中著名的龙格-库塔格式。在所提出的方法中，由于分数阶导数依赖于固定基点t0，我们必须在FORK方法的所有步骤中修改给定方程的右端项。本文给出了显式和隐式FORK格式的一些系数，并讨论了所提出方法的收敛性分析。数值实验验证了该方法的有效性和鲁棒性。