Math. Model. Nat. Phenom.
Volume 13, Number 1, 2018
Theory and applications of fractional differentiation
|Number of page(s)||17|
|Published online||26 February 2018|
Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State,
9300, South Africa
2 Department of Mathematical Sciences, Federal University of Technology, PMB 704, Akure, Ondo State, Nigeria
* Corresponding author: firstname.lastname@example.org
Accepted: 27 September 2017
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.
Mathematics Subject Classification: 26A33 / 33E12 / 34A34 / 35B44 / 65M06
Key words: Atangana-Baleanu derivative / bifurcation analysis / blow-up process / chaotic and spatiotemporal oscillations / Mittag-Leffler law
© EDP Sciences, 2018
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