Abstract
This paper is devoted to solving the initial value problem (IVP) of the fractional differential equation (FDE) in Caputo sense for arbitrary order $ \beta\in(0, 1] $. Based on a few examples and application models, the main motivation is to show that FDE may model more effectively than the ordinary differential equation (ODE). Here, two cubic convergence numerical schemes are developed: the fractional third-order Runge-Kutta (RK3) scheme and fractional strong stability preserving third-order Runge-Kutta (SSRK3) scheme. The approximated solution is derived without taking any assumption of perturbations and linearization. The schemes are presented, and the convergence of the schemes is established. Also, a comparative study has been done of our proposed scheme with fractional Euler method (EM) and fractional improved Euler method (IEM), which has linear and quadratic convergence rates, respectively. Illustrative examples and application examples with the numerical comparison between the proposed scheme, the exact solution, EM, and IEM are given to reveal our scheme's accuracy and efficiency.