Fractional Order Runge–Kutta Methods

Abstract

This paper presents a new class of fractional order Runge–Kutta (FORK) methods for numerically approximating the solution of fractional differential equations (FDEs). We construct explicit and implicit FORK methods for FDEs by using the Caputo generalized Taylor series formula. Due to the dependence of fractional derivatives on a fixed base point, in the proposed method, 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 are presented to clarify the effectiveness and robustness of the method.

Solvability and Ulam-Hyers stability analysis for nonlinear piecewise fractional cancer dynamic systems

“…We also create a plan for our numerical results using certain criteria for definite integrals. First, develop the approach for the system's first equation (1), then repeat the same techniques for the next equations, details in [42]. The proposed scheme is also given for simple problem.…”

“…Further, the existence theory of solution is important area to investigate before computing numerical solution to the proposed problem. For the importance of numerical analysis of fractional order problems, we refer some work as [42,43].…”

Mittag-Leffler Functions in the Integral Representation of a Class of FDEs: Existence, Stability and Numirical Study.

“…In special cases, the Runge-Kutta method is considered a good option to solve fractional differential equations. This method received great attention from researchers; for more information, the reader can see the works [16], [17], [31], [35]. Also, the quasi-linearisation method is considered a good technique for giving the approximate solutions to fractional differential equations, we cite [2], [11], [37].…”