Generalised forms of fractional Euler and Runge-Kutta methods using non-uniform grid

Abstract

In this paper, we propose generalised forms of three well-known fractional numerical methods named as Euler, Runge-Kutta 2-step, and Runge-Kutta 4-step, respectively. The new versions we provide of these methods are derived by using a technique of non-uniform grid which is slightly di erent to all the previous existing versions of these algorithms. A new generalised form of well-known Caputo-type fractional derivative is used to derive the results. All necessary analysis related to the stability, convergence, and error bounds are also provided. The precision of all simulated results are justi ed by performing a bunch of practical works in which some important examples have been solved by implementing the code in Mathematica software. Finally, we give a brief discussion on the simulated results which shows that the generalised methods are fully novel, e ective, strong, and very easy to implement.