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

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.