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

In this article, we propose generalized forms of three well-known fractional numerical methods namely Euler, Runge-Kutta 2-step, and Runge-Kutta 4-step, respectively. The new versions we provide of these methods are derived by utilizing a non-uniform grid which is slightly different from previous versions of these algorithms. A new generalized form of the well-known Caputo-type fractional derivative is used to derive the results. All necessary analyses related to the stability, convergence, and error bounds are also provided. The precision of all simulated results is justified by performing multiple numerical experiments, with some meaningful problems solved by implementing the code in Mathematica. Finally, we give a brief discussion on the simulated results which shows that the generalized methods are novel, effective, reliable, and very easy to implement.