Numerical Schemes for the Fractional-Order Logistic Growth Model with Caputo-Fabrizio Operator

In this paper, we aim to examine and compare some numerical schemes for the fractional-order logistic growth model with respect to the Caputo-Fabrizio operator. We provide some numerical schemes of the solution using linear interpolation (LLI), three-point centered interpolation (LCI), three-point backward interpolation (LBI), and three-point extrapolation (LE), which are Lagrange-based approximations. An LCI-based scheme is a new scheme, and the other schemes are derived from previous studies. We perform all numerical schemes to model for different values of step sizes and derivative orders and then compare them to the analytical solution. The analytical solution of the model was obtained via the corresponding fractional-order integral. We determine the root mean square error (RMSE) for each numerical scheme. The LBI-based scheme generally has the best performance in all conditions, with the smallest RMSE. The LE-based scheme has good performance for small steps. However, it is disreputable for its quite large step size, even providing spikes and an unsmooth solution. This leads to a change in the convergence of solutions. Finally, two other schemes have consistent performance in all conditions of step size, where the LCI-based scheme is slightly better than the LLI-based scheme. For an approximation of the analytical solution, we suggest using the LBI-based scheme for the best results.