Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation
The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/381753 |
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| author | Abdon Atangana P. D. Vermeulen |
| author_facet | Abdon Atangana P. D. Vermeulen |
| author_sort | Abdon Atangana |
| collection | DOAJ |
| description | The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given. |
| format | Article |
| id | doaj-art-f0ee7a2dfe2d4185b3307c1390e3bc25 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-f0ee7a2dfe2d4185b3307c1390e3bc252025-02-03T01:04:43ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/381753381753Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow EquationAbdon Atangana0P. D. Vermeulen1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 399, Bloemfontein, South AfricaInstitute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, P.O. Box 399, Bloemfontein, South AfricaThe classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given.http://dx.doi.org/10.1155/2014/381753 |
| spellingShingle | Abdon Atangana P. D. Vermeulen Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation Abstract and Applied Analysis |
| title | Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation |
| title_full | Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation |
| title_fullStr | Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation |
| title_full_unstemmed | Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation |
| title_short | Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation |
| title_sort | analytical solutions of a space time fractional derivative of groundwater flow equation |
| url | http://dx.doi.org/10.1155/2014/381753 |
| work_keys_str_mv | AT abdonatangana analyticalsolutionsofaspacetimefractionalderivativeofgroundwaterflowequation AT pdvermeulen analyticalsolutionsofaspacetimefractionalderivativeofgroundwaterflowequation |