Modelling and analysis of fractal-fractional partial differential equations: Application to reaction-diffusion model

In this paper, an extension is paid to an idea of fractal and fractional derivatives which has been applied to a number of ordinary differential equations to model a system of partial differential equations. As a case study, the fractal fractional Schnakenberg system is formulated with the Caputo op...

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Bibliographic Details
Published in:Alexandria engineering journal 2020-08, Vol.59 (4), p.2477-2490
Main Authors: Owolabi, Kolade M., Atangana, Abdon, Akgul, Ali
Format: Article
Language:English
Subjects:
34A34
35A05
35K57
65L05
65M06
93C10
Caputo and Caputo-Fabrizio derivatives
Fractal operator
Fractional reaction-diffusion
Linear stability analysis
Mittag-Leffler kernel
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Summary:In this paper, an extension is paid to an idea of fractal and fractional derivatives which has been applied to a number of ordinary differential equations to model a system of partial differential equations. As a case study, the fractal fractional Schnakenberg system is formulated with the Caputo operator (in terms of the power law), the Caputo-Fabrizio operator (with exponential decay law) and the Atangana-Baleanu fractional derivative (based on the Mittag-Liffler law). We design some algorithms for the Schnakenberg model by using the newly proposed numerical methods. In such schemes, it worth mentioning that the classical cases are recovered whenever α=1 and β=1. Numerical results obtained for different fractal-order (β∈(0,1)) and fractional-order (α∈(0,1)) are also given to address any point and query that may arise.
ISSN:1110-0168
DOI:10.1016/j.aej.2020.03.022