Title: The Caputo-Fabrizio fractional derivative applied to a singular perturbation problem

Authors: Abdon Atangana; Emile Franc Doungmo Goufo

Addresses: Institute for Groundwater Studies, University of the Free State, Bloemfontein, 9300 South Africa ' Department of Mathematical Sciences, University of South Africa, Florida, 0003 South Africa

Abstract: The garden equation is a nonlinear partial differential equation that has application in more than two different fields. In this paper, we use the Caputo-Fabrizio derivative with fractional order to extend this model to the concept of fractional calculus. In the process, we prove that the new derivative satisfies the equality of mixed partial and in the extended equation, we present the analysis of existence and uniqueness of the exact solution. We propose a special solution using the Laplace iterative methods. Some numerical simulations are preformed for different values of alpha and also the perturbed parameter.

Keywords: nonlinear garden equation; Caputo-Fabrizio fractional derivative; equality of mixed partial; existence and uniqueness; numerical solutions.

DOI: 10.1504/IJMMNO.2019.100486

International Journal of Mathematical Modelling and Numerical Optimisation, 2019 Vol.9 No.3, pp.241 - 253

Received: 06 Apr 2018
Accepted: 05 Jun 2018

Published online: 29 Jun 2019 *

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