Fractional Reaction-Diffusion Equations for Modelling Complex Biological Patterns

Ku Azlina Ku Akil,Einly Lim,Sithi V Muniandy

doi:10.11113/mjfas.v8n3.135

Ku Azlina Ku Akil, Einly Lim + Show 1 more

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https://doi.org/10.11113/mjfas.v8n3.135

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Abstract

We consider a system of nonlinear time-fractional reaction-diffusion equations (TFRDE) on a finite spatial domain x ∈ [0, L], and time t ∈ [0, T]. The system of standard reaction-diffusion equations with Neumann boundary conditions are generalized by replacing the first-order time derivatives with Caputo time-fractional derivatives of order α ∈ (0, 1). We solve the TFRDE numerically using Grünwald-Letnikov derivative approximation for time-fractional derivative and centred difference approximation for spatial derivative. We discuss the numerical results and propose the applications of TFRDE for the modelling of complex patterns in biological systems.

Highlights

The system of fractional reactiondiffusion (FRD) equations have gained considerable popularity to study nonlinear phenomena arise in the disciplines of science and engineering [1]
The evolution of pattern formation is best described by the fractional-order models because the fractional derivatives take into consideration the whole history of the system which is called the memory effect [7]
The fractional reaction-diffusion equation is obtained by replacing the first-order time derivative index by α ∈ (0,1), or the second-order spatial derivative index by β ∈ (1,2), or both in the standard reaction-diffusion equation, (1)

Summary

INTRODUCTION

The system of fractional reactiondiffusion (FRD) equations have gained considerable popularity to study nonlinear phenomena arise in the disciplines of science and engineering [1]. Of these particular interests are patterns formations [2,3,4,5,6]. The numerical approaches for approximating solutions to fractional reaction-diffusion equations have been widely studied. In this paper we consider a o ne-dimensional system of nonlinear time-fractional reaction-diffusion equations (TFRDE) on a finite spatial domain and time with Neumann boundary conditions. Akil et al / Malaysian Journal of Fundamental & Applied Sciences Vol., No.3 (2012) 126-130

MATHEMATICAL MODEL

NUMERICAL SCHEME and

RESULTS & DISCUSSION

CONCLUSION