A numerical study on fractional differential equation with population growth model

In this work, we developed two efficient and fast numerical technique to solve an initial value problem (IVP) of the linear and nonlinear fractional differential equations (FDEs) of order α, 0 < α < 1. Here we have used the arbitrary order derivatives in Riemann style. The proposed algorithm are very accurate and provides the solutions directly without perturbations, linearization, or any other assumptions. Illustrating examples with numerical comparisons between the proposed algorithm and the exact and/or Euler method and the improved Euler method (IEM) are given to reveal the efficiency and the accuracy of our algorithm. These scheme has quadratic and cubic convergence rate which is faster than the Euler method and IEM for solving the IVP of FDEs. Moreover, we have discussed the behaviors through graphical representation of the obtained solutions. Furthermore, both methods will be useful for the treatment of disease models for further study.