Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense

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

^* Corresponding author: mkowolax@yahoo.com

Received: 26 September 2017
Accepted: 27 September 2017

Abstract

In this paper, we consider a numerical approach for fourth-order time fractional partial differential equation. This equation is obtained from the classical reaction-diffusion equation by replacing the first-order time derivative with the Atangana-Baleanu fractional derivative in Riemann-Liouville sense with the Mittag-Leffler law kernel, and the first, second, and fourth order space derivatives with the fourth-order central difference schemes. We also suggest the Fourier spectral method as an alternate approach to finite difference. We employ Plais Fourier method to study the question of finite-time singularity formation in the one-dimensional problem on a periodic domain. Our bifurcation analysis result shows the relationship between the blow-up and stability of the steady periodic solutions. Numerical experiments are given to validate the effectiveness of the proposed methods.