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

Sami Aljhani,A K Alomari,Mohd Salmi Md Noorani,Khaled M Saad,Dumitru Vieru

doi:10.1155/2021/2544688

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 ℏ 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.

Highlights

Differential equations play a significant role within the field of finance, engineering, physics, and biology
To demonstrate the efficiency of the fractional homotopy analysis transform method (FHATM) for solving the time-fractional Gray-Scott equation, we present the solution in figures and tables for several values of fractional derivatives
We evaluated the intervals of convergence for the LC, CFC, and ABC by finding ħu,v curves, and the averaged residual error

Summary

Introduction

Differential equations play a significant role within the field of finance, engineering, physics, and biology. Thereafter, Atangana and Baleanu (AB) developed a new concept of differentiation with nonsingular [25, 26], based on the general Mittag-Leffler function These two concepts with fractional order in RiemannLiouville and Liouville-Caputo sense have a nonlocal kernel. Khan et al [32] and Kumar et al [33, 34] coupled the homotopy analysis method (HAM) [35,36,37] with the Laplace transform to solve a nonlinear differential equation. This method is called the fractional homotopy analysis transform method (FHATM). To the best of our knowledge, this paper is the first one that introduced the approximate analytic solution for the time-fractional Gray-Scott system using a nonsingular fractional derivative

Preliminaries and Notations

Homotopy and Laplace Transform for FHATM

FðηÞ ηÞ ηρÞ

Numerical Results

Conclusion