Eur. Phys. J. Plus (2018) 133: 98
https://doi.org/10.1140/epjp/i2018-11951-x

Regular Article

Efficient numerical simulation of non-integer-order space-fractional reaction-diffusion equation via the Riemann-Liouville operator

¹ Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria
² Institute for Groundwater Studies, Faculty of Natural, Agricultural Sciences University of the Free State, 9300, Bloemfontein, South Africa

^* e-mail: kmowolabi@futa.edu.ng

Received: 19 January 2018
Accepted: 9 February 2018
Published online: 8 March 2018

Abstract

In this work, we are concerned with the solution of non-integer space-fractional reaction-diffusion equations with the Riemann-Liouville space-fractional derivative in high dimensions. We approximate the Riemann-Liouville derivative with the Fourier transform method and advance the resulting system in time with any time-stepping solver. In the numerical experiments, we expect the travelling wave to arise from the given initial condition on the computational domain , which we terminate in the numerical experiments with a large but truncated value of L. It is necessary to choose L large enough to allow the waves to have enough space to distribute. Experimental results in high dimensions on the space-fractional reaction-diffusion models with applications to biological models (Fisher and Allen-Cahn equations) are considered. Simulation results reveal that fractional reaction-diffusion equations can give rise to a range of physical phenomena when compared to non-integer-order cases. As a result, most meaningful and practical situations are found to be modelled with the concept of fractional calculus.