A study of Ralston's cubic convergence with the application of population growth model

This paper deals a new numerical scheme to solve fractional differential equation (FDE) involving Caputo fractional derivative (CFD) of variable order $ \beta \in ]0, 1] $. Based on a few examples and application models, the main objective is to show that FDE works more effectively than ordinary differential equations (ODEs). The proposed scheme is fractional Ralston's cubic method (RCM). The convergence analysis and stability analysis of the scheme is proved. The numerical scheme has been found without considering linearisation, perturbations, or any such assumptions. Finally, the efficiency of the proposed scheme will justify by solving a few examples of linear and non-linear FDEs with one application of FDE, world population growth (WPG) model of variable order $ \beta \in ]0, 1] $. Also, the comparison of fractional RCM scheme has been shown with the existing fractional Euler method (EM) and fractional improved Euler method (IEM).