Numerical Solutions of Certain New Models of the Time-Fractional Gray-Scott

Abstract

A reaction-diffusion system can be represented by the Gray-Scott model. In this study, we discuss a one-dimensional time-fractional Gray-Scott model with Liouville-Caputo, Caputo-Fabrizio-Caputo, and Atangana-Baleanu-Caputo fractional derivatives. We utilize the fractional homotopy analysis transformation method to obtain approximate solutions for the time-fractional Gray-Scott model. This method gives a more realistic series of solutions that converge rapidly to the exact solution. We can ensure convergence by solving the series resultant. We study the convergence analysis of fractional homotopy analysis transformation method by determining the interval of convergence employing the ${\hslash }_{u,v}$ -curves and the average residual error. We also test the accuracy and the efficiency of this method by comparing our results numerically with the exact solution. Moreover, the effect of the fractionally obtained derivatives on the reaction-diffusion is analyzed. The fractional homotopy analysis transformation method algorithm can be easily applied for singular and nonsingular fractional derivative with partial differential equations, where a few terms of series solution are good enough to give an accurate solution.