Open Access Research

A generalized groundwater flow equation using the concept of variable-order derivative

Abdon Atangana* and Joseph Francois Botha

Author affiliations

Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 9301, Bloemfontein, South Africa

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Citation and License

Boundary Value Problems 2013, 2013:53  doi:10.1186/1687-2770-2013-53

Published: 14 March 2013

Abstract

In this paper, the groundwater flow equation is generalized using the concept of the variational order derivative. We present a numerical solution of the modified groundwater flow equation with the variational order derivative. We solve the generalized equation with the Crank-Nicholson technique. Numerical methods typically yield approximate solutions to the governing equation through the discretization of space and time and can relax the rigid idealized conditions of analytical models or lumped-parameter models. They can therefore be more realistic and flexible for simulating field conditions. Within the discredited problem domain, the variable internal properties, boundaries, and stresses of the system are approximated. We perform the stability and convergence analysis of the Crank-Nicholson method and complete the paper with some illustrative computational examples and their simulations.

Keywords:
groundwater flow equation; variable order derivative; Crank-Nicholson scheme; stability; convergence