# American Institute of Mathematical Sciences

July  2019, 24(7): 3227-3247. doi: 10.3934/dcdsb.2018317

## A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative

 1 African Institute for Mathematical Sciences-Cameroon, Limbe Crystal Gardens, South West Region, P.O. Box 608, Cameroon 2 Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of Free Staye, Bloemfontein, 9300, South Africa

* Corresponding author: Atangana Abdon

Received  July 2017 Revised  March 2018 Published  January 2019

Fund Project: The first author was supported by AIMS-Cameroon Scholarship grant 2015-2016.
The second author was supported by AIMS-Cameroon tutor fellowship grant 2015-2016.

In this paper, the groundwater flow equation within an unconfined aquifer is modified using the concept of new derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. Some properties and applications are given regarding the Caputo-Fabrizio fractional order derivative. The existence and the uniqueness of the solution of the modified groundwater flow equation within an unconfined aquifer is presented, the proof of the existence use the definition of Caputo-Fabrizio integral and the powerful fixed-point Theorem. A detailed analysis on the uniqueness is included. We perform on the numerical analysis on which the Crank-Nicolson scheme is used for discretisation. Then we present in particular the proof of the stability of the method, the proof combine the Fourier and Von Neumann stability analysis. A detailed analysis on the convergence is also achieved.

Citation: Pierre Aime Feulefack, Jean Daniel Djida, Atangana Abdon. A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3227-3247. doi: 10.3934/dcdsb.2018317
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