A novel time efficient structure-preserving splitting method for the solution of two-dimensional reaction-diffusion systems

Nauman Ahmed,M S Iqbal,Ali Saleh Alshomrani,Alper Korkmaz,M A Rehman,Dumitru Baleanu,M Rafiq

doi:10.1186/s13662-020-02659-0

Abstract

In this article, the first part is concerned with the important questions related to the existence and uniqueness of solutions for nonlinear reaction-diffusion systems. Secondly, an efficient positivity-preserving operator splitting nonstandard finite difference scheme (NSFD) is designed for such a class of systems. The presented formulation is unconditionally stable as well as implicit in nature and even time efficient. The proposed NSFD operator splitting technique also preserves all the important properties possessed by continuous systems like positivity, convergence to the fixed points of the system, and boundedness. The proposed algorithm is implicit in nature but more efficient in time than the extensively used Euler method.

Highlights

Reaction-diffusion equations generally arise in different chemical and biological models which describe the physical phenomena like concentration, density, population sizes, and many more
The nonstandard finite difference scheme (NSFD) method proposed by Mickens [1] is an efficient way to design structure-preserving finite difference (FD) schemes
We propose the positivity preserving NSFD operator splitting technique for the solution of two-dimensional reaction-diffusion systems

Summary

Introduction

Reaction-diffusion equations generally arise in different chemical and biological models which describe the physical phenomena like concentration, density, population sizes, and many more. Various writers proposed NSFD and positivity preserving FD schemes for the solution of differential equations. Many authors used the NSFD technique to find the solution of ordinary differential equations arising in chemical and biological models, for the readers, some of the references are presented [2,3,4,5,6,7,8]. Different NSFD and positivity preserving FD methods are introduced in the literature to solve reactiondiffusion equations [9,10,11,12,13,14,15,16,17]. Various techniques are used to solve a similar type of diffusion systems [18,19,20,21,22,23]

Objectives

Methods

Conclusion