# 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
##### References:
 [1] R. T. Alqahtani, Fixed-point theorem for Caputo-Fabrizio fractional Nagumo equation with nonlinear diffusion and convection, in J. Nonlinear Sci. Appl, 9 (2016), 1991-1999. doi: 10.22436/jnsa.009.05.05. [2] A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: Application of Caputo-Fabrizio derivative, in Arabian Journal of Geosciences, Springer, 9 (2016), 8pp. [3] A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, in Entropy, Multidisciplinary Digital Publishing Institute, 17 (2015), 4439-4453. doi: 10.3390/e17064439. [4] A. Atangana and N. Bildik, The use of fractional order derivative to predict the groundwater flow, in Hindawi Publishing Corporation, Mathematical Problems in Engineering, 2013 (2013), Art. ID 543026, 9 pp. doi: 10.1155/2013/543026. [5] A. Atangana and P. D. Vermeulen, Analytical solutions of a space-time fractional derivative of groundwater flow equation, in Hindawi, 2014 (2014), Art. ID 381753, 11 pp. doi: 10.1155/2014/381753. [6] A. Atangana and J. F. Botha, A generalized groundwater flow equation using the concept of variable-order derivative, in Boundary Value Problems, Springer, 2013 (2013), 1-11. doi: 10.1186/1687-2770-2013-53. [7] A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, in Advances in Mechanical Engineering, SAGE Publications 7 (2015), 1687814015613758. [8] N. S. Boulton, Unsteady radial flow to a pumped well allowing for delayed yield from storage, in Int. Assoc. Sci. Hydrol. Publ, 2 (1954), 472-477. [9] H. Brezis, Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, in Progr. Fract. Differ. Appl, 1 (2015), 1-13. [11] C.-M. Chen, et al, A Fourier method for the fractional diffusion equation describing subdiffusion, in Journal of Computational Physics, 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012. [12] C.-M. Chen, et al, Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation, in Mathematics of Computation, 81 (2012), 345-366. doi: 10.1090/S0025-5718-2011-02447-6. [13] A. Cloot and J. F. Botha, A generalised groundwater flow equation using the concept of non-integer order derivatives, in Water SA, Water Research Commission (WRC), 32 (2007), 1-7. [14] K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, in Numerical algorithms, Springer, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be. [15] Eng. Deeb Abdel-Ghafour, Pumping test for groundwater aquifers analysis and evaluation, 2005, available from: https://docplayer.net/11404875-Pumping-test-for-groundwater-aquifers-analysis-and-evaluation-by-eng-deeb-abdel-ghafour.html. [16] G. Gambolati, Analytic element modeling of groundwater flow, in Eos, Transactions Ameriocan Geophysical Union, 77 (1995), 103-103. [17] G. Garven and R. A. Freeze, Theoretical analysis of the role of groundwater flow in the genesis of stratabound ore deposits, in Mathematical and Numerical Model, American Journal of Science, 284 (1984), 1085-1124. [18] H. M. Haitjema, Analytic element modeling of groundwater flow, in nc San Diego, CA, USA Google Scholar, Academic Press, (1995), 33-75. [19] L. F. Konikow and D. B. Grove, Derivation of equations describing solute transport in ground water, in US Geological Survey, Water Resources Division, 77 (1977). [20] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, in Progr. Fract. Differ. Appl, 1 (2015), 87-92. [21] P. K. Mishra and K. L. Kuhlman, Unconfined aquifer flow theory: from Dupuit to present, in Advances in Hydrogeology, Springer, New York, NY (2013), 185-202. [22] Pollock and W. David, Documentation of computer programs to compute and display pathlines using results from the US Geological Survey modular three-dimensional finite-difference ground-water flow model, in US Geological Survey, 89 (1989). [23] J. R. Prendergast, R. M. Quinn and J. H. Lawton, The gaps between theory and practice in selecting nature reserves, Conservation Biology, Wiley Online Library, 13 (1999), 484-492. [24] S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-68093-2. [25] C. V. Theis, The relation between the lowering of the Piezometric surface and the rate and duration of discharge of a well using ground-water storage, in Eos, Transactions American Geophysical Union, Wiley Online Library, 16 (1935), 519-524. [26] G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, in Integrated Education, TaylorFrancis, 24 (1993), 35-43. doi: 10.1080/0020739930240105. [27] S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, in Journal on Numerical Analysis, SIAM, 42 (2005), 1862-1874. doi: 10.1137/030602666. [28] I. S. Zektser, E. Lorne and others, Groundwater Resources of the World: And Their Use, IhP Series on groundwater, 6nd edition, Unesco, 2004.

show all references

##### References:
 [1] R. T. Alqahtani, Fixed-point theorem for Caputo-Fabrizio fractional Nagumo equation with nonlinear diffusion and convection, in J. Nonlinear Sci. Appl, 9 (2016), 1991-1999. doi: 10.22436/jnsa.009.05.05. [2] A. Atangana and B. S. T. Alkahtani, New model of groundwater flowing within a confine aquifer: Application of Caputo-Fabrizio derivative, in Arabian Journal of Geosciences, Springer, 9 (2016), 8pp. [3] A. Atangana and B. S. T. Alkahtani, Analysis of the Keller-Segel model with a fractional derivative without singular kernel, in Entropy, Multidisciplinary Digital Publishing Institute, 17 (2015), 4439-4453. doi: 10.3390/e17064439. [4] A. Atangana and N. Bildik, The use of fractional order derivative to predict the groundwater flow, in Hindawi Publishing Corporation, Mathematical Problems in Engineering, 2013 (2013), Art. ID 543026, 9 pp. doi: 10.1155/2013/543026. [5] A. Atangana and P. D. Vermeulen, Analytical solutions of a space-time fractional derivative of groundwater flow equation, in Hindawi, 2014 (2014), Art. ID 381753, 11 pp. doi: 10.1155/2014/381753. [6] A. Atangana and J. F. Botha, A generalized groundwater flow equation using the concept of variable-order derivative, in Boundary Value Problems, Springer, 2013 (2013), 1-11. doi: 10.1186/1687-2770-2013-53. [7] A. Atangana and J. J. Nieto, Numerical solution for the model of RLC circuit via the fractional derivative without singular kernel, in Advances in Mechanical Engineering, SAGE Publications 7 (2015), 1687814015613758. [8] N. S. Boulton, Unsteady radial flow to a pumped well allowing for delayed yield from storage, in Int. Assoc. Sci. Hydrol. Publ, 2 (1954), 472-477. [9] H. Brezis, Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [10] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, in Progr. Fract. Differ. Appl, 1 (2015), 1-13. [11] C.-M. Chen, et al, A Fourier method for the fractional diffusion equation describing subdiffusion, in Journal of Computational Physics, 227 (2007), 886-897. doi: 10.1016/j.jcp.2007.05.012. [12] C.-M. Chen, et al, Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation, in Mathematics of Computation, 81 (2012), 345-366. doi: 10.1090/S0025-5718-2011-02447-6. [13] A. Cloot and J. F. Botha, A generalised groundwater flow equation using the concept of non-integer order derivatives, in Water SA, Water Research Commission (WRC), 32 (2007), 1-7. [14] K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, in Numerical algorithms, Springer, 36 (2004), 31-52. doi: 10.1023/B:NUMA.0000027736.85078.be. [15] Eng. Deeb Abdel-Ghafour, Pumping test for groundwater aquifers analysis and evaluation, 2005, available from: https://docplayer.net/11404875-Pumping-test-for-groundwater-aquifers-analysis-and-evaluation-by-eng-deeb-abdel-ghafour.html. [16] G. Gambolati, Analytic element modeling of groundwater flow, in Eos, Transactions Ameriocan Geophysical Union, 77 (1995), 103-103. [17] G. Garven and R. A. Freeze, Theoretical analysis of the role of groundwater flow in the genesis of stratabound ore deposits, in Mathematical and Numerical Model, American Journal of Science, 284 (1984), 1085-1124. [18] H. M. Haitjema, Analytic element modeling of groundwater flow, in nc San Diego, CA, USA Google Scholar, Academic Press, (1995), 33-75. [19] L. F. Konikow and D. B. Grove, Derivation of equations describing solute transport in ground water, in US Geological Survey, Water Resources Division, 77 (1977). [20] J. Losada and J. J. Nieto, Properties of a new fractional derivative without singular kernel, in Progr. Fract. Differ. Appl, 1 (2015), 87-92. [21] P. K. Mishra and K. L. Kuhlman, Unconfined aquifer flow theory: from Dupuit to present, in Advances in Hydrogeology, Springer, New York, NY (2013), 185-202. [22] Pollock and W. David, Documentation of computer programs to compute and display pathlines using results from the US Geological Survey modular three-dimensional finite-difference ground-water flow model, in US Geological Survey, 89 (1989). [23] J. R. Prendergast, R. M. Quinn and J. H. Lawton, The gaps between theory and practice in selecting nature reserves, Conservation Biology, Wiley Online Library, 13 (1999), 484-492. [24] S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-540-68093-2. [25] C. V. Theis, The relation between the lowering of the Piezometric surface and the rate and duration of discharge of a well using ground-water storage, in Eos, Transactions American Geophysical Union, Wiley Online Library, 16 (1935), 519-524. [26] G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, in Integrated Education, TaylorFrancis, 24 (1993), 35-43. doi: 10.1080/0020739930240105. [27] S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, in Journal on Numerical Analysis, SIAM, 42 (2005), 1862-1874. doi: 10.1137/030602666. [28] I. S. Zektser, E. Lorne and others, Groundwater Resources of the World: And Their Use, IhP Series on groundwater, 6nd edition, Unesco, 2004.
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