
Previous Article
Polynomial maps with hidden complex dynamics
 DCDSB Home
 This Issue

Next Article
Pointwise wave behavior of the initialboundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $
A new model of groundwater flow within an unconfined aquifer: Application of CaputoFabrizio fractional derivative
1.  African Institute for Mathematical SciencesCameroon, 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 
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 CaputoFabrizio 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 CaputoFabrizio integral and the powerful fixedpoint Theorem. A detailed analysis on the uniqueness is included. We perform on the numerical analysis on which the CrankNicolson 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.
References:
[1] 
R. T. Alqahtani, Fixedpoint theorem for CaputoFabrizio fractional Nagumo equation with nonlinear diffusion and convection, in J. Nonlinear Sci. Appl, 9 (2016), 19911999. 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 CaputoFabrizio derivative, in Arabian Journal of Geosciences, Springer, 9 (2016), 8pp. 
[3] 
A. Atangana and B. S. T. Alkahtani, Analysis of the KellerSegel model with a fractional derivative without singular kernel, in Entropy, Multidisciplinary Digital Publishing Institute, 17 (2015), 44394453. 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 spacetime 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 variableorder derivative, in Boundary Value Problems, Springer, 2013 (2013), 111. doi: 10.1186/16872770201353. 
[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), 472477. 
[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), 113. 
[11] 
C.M. Chen, et al, A Fourier method for the fractional diffusion equation describing subdiffusion, in Journal of Computational Physics, 227 (2007), 886897. doi: 10.1016/j.jcp.2007.05.012. 
[12] 
C.M. Chen, et al, Numerical methods for solving a twodimensional variableorder anomalous subdiffusion equation, in Mathematics of Computation, 81 (2012), 345366. doi: 10.1090/S002557182011024476. 
[13] 
A. Cloot and J. F. Botha, A generalised groundwater flow equation using the concept of noninteger order derivatives, in Water SA, Water Research Commission (WRC), 32 (2007), 17. 
[14] 
K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, in Numerical algorithms, Springer, 36 (2004), 3152. doi: 10.1023/B:NUMA.0000027736.85078.be. 
[15] 
Eng. Deeb AbdelGhafour, Pumping test for groundwater aquifers analysis and evaluation, 2005, available from: https://docplayer.net/11404875Pumpingtestforgroundwateraquifersanalysisandevaluationbyengdeebabdelghafour.html. 
[16] 
G. Gambolati, Analytic element modeling of groundwater flow, in Eos, Transactions Ameriocan Geophysical Union, 77 (1995), 103103. 
[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), 10851124. 
[18] 
H. M. Haitjema, Analytic element modeling of groundwater flow, in nc San Diego, CA, USA Google Scholar, Academic Press, (1995), 3375. 
[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), 8792. 
[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), 185202. 
[22] 
Pollock and W. David, Documentation of computer programs to compute and display pathlines using results from the US Geological Survey modular threedimensional finitedifference groundwater 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), 484492. 
[24] 
S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. SpringerVerlag, Berlin, 2011. doi: 10.1007/9783540680932. 
[25] 
C. V. Theis, The relation between the lowering of the Piezometric surface and the rate and duration of discharge of a well using groundwater storage, in Eos, Transactions American Geophysical Union, Wiley Online Library, 16 (1935), 519524. 
[26] 
G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, in Integrated Education, TaylorFrancis, 24 (1993), 3543. doi: 10.1080/0020739930240105. 
[27] 
S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumanntype stability analysis for fractional diffusion equations, in Journal on Numerical Analysis, SIAM, 42 (2005), 18621874. doi: 10.1137/030602666. 
[28] 
I. S. Zektser, E. Lorne and others, Groundwater Resources of the World: And Their Use, IhP Series on groundwater, 6^{nd} edition, Unesco, 2004. 
show all references
References:
[1] 
R. T. Alqahtani, Fixedpoint theorem for CaputoFabrizio fractional Nagumo equation with nonlinear diffusion and convection, in J. Nonlinear Sci. Appl, 9 (2016), 19911999. 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 CaputoFabrizio derivative, in Arabian Journal of Geosciences, Springer, 9 (2016), 8pp. 
[3] 
A. Atangana and B. S. T. Alkahtani, Analysis of the KellerSegel model with a fractional derivative without singular kernel, in Entropy, Multidisciplinary Digital Publishing Institute, 17 (2015), 44394453. 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 spacetime 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 variableorder derivative, in Boundary Value Problems, Springer, 2013 (2013), 111. doi: 10.1186/16872770201353. 
[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), 472477. 
[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), 113. 
[11] 
C.M. Chen, et al, A Fourier method for the fractional diffusion equation describing subdiffusion, in Journal of Computational Physics, 227 (2007), 886897. doi: 10.1016/j.jcp.2007.05.012. 
[12] 
C.M. Chen, et al, Numerical methods for solving a twodimensional variableorder anomalous subdiffusion equation, in Mathematics of Computation, 81 (2012), 345366. doi: 10.1090/S002557182011024476. 
[13] 
A. Cloot and J. F. Botha, A generalised groundwater flow equation using the concept of noninteger order derivatives, in Water SA, Water Research Commission (WRC), 32 (2007), 17. 
[14] 
K. Diethelm, N. J. Ford and A. D. Freed, Detailed error analysis for a fractional Adams method, in Numerical algorithms, Springer, 36 (2004), 3152. doi: 10.1023/B:NUMA.0000027736.85078.be. 
[15] 
Eng. Deeb AbdelGhafour, Pumping test for groundwater aquifers analysis and evaluation, 2005, available from: https://docplayer.net/11404875Pumpingtestforgroundwateraquifersanalysisandevaluationbyengdeebabdelghafour.html. 
[16] 
G. Gambolati, Analytic element modeling of groundwater flow, in Eos, Transactions Ameriocan Geophysical Union, 77 (1995), 103103. 
[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), 10851124. 
[18] 
H. M. Haitjema, Analytic element modeling of groundwater flow, in nc San Diego, CA, USA Google Scholar, Academic Press, (1995), 3375. 
[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), 8792. 
[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), 185202. 
[22] 
Pollock and W. David, Documentation of computer programs to compute and display pathlines using results from the US Geological Survey modular threedimensional finitedifference groundwater 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), 484492. 
[24] 
S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Series in Computational Mathematics, 39. SpringerVerlag, Berlin, 2011. doi: 10.1007/9783540680932. 
[25] 
C. V. Theis, The relation between the lowering of the Piezometric surface and the rate and duration of discharge of a well using groundwater storage, in Eos, Transactions American Geophysical Union, Wiley Online Library, 16 (1935), 519524. 
[26] 
G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, in Integrated Education, TaylorFrancis, 24 (1993), 3543. doi: 10.1080/0020739930240105. 
[27] 
S. B. Yuste and L. Acedo, An explicit finite difference method and a new von Neumanntype stability analysis for fractional diffusion equations, in Journal on Numerical Analysis, SIAM, 42 (2005), 18621874. doi: 10.1137/030602666. 
[28] 
I. S. Zektser, E. Lorne and others, Groundwater Resources of the World: And Their Use, IhP Series on groundwater, 6^{nd} edition, Unesco, 2004. 
[1] 
Sondre Tesdal Galtung. A convergent CrankNicolson Galerkin scheme for the BenjaminOno equation. Discrete & Continuous Dynamical Systems  A, 2018, 38 (3) : 12431268. doi: 10.3934/dcds.2018051 
[2] 
Yingwen Guo, Yinnian He. Fully discrete finite element method based on secondorder CrankNicolson/AdamsBashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems  B, 2015, 20 (8) : 25832609. doi: 10.3934/dcdsb.2015.20.2583 
[3] 
Dongho Kim, EunJae Park. Adaptive CrankNicolson methods with dynamic finiteelement spaces for parabolic problems. Discrete & Continuous Dynamical Systems  B, 2008, 10 (4) : 873886. doi: 10.3934/dcdsb.2008.10.873 
[4] 
Alexander Zlotnik. The NumerovCrankNicolson scheme on a nonuniform mesh for the timedependent Schrödinger equation on the halfaxis. Kinetic & Related Models, 2015, 8 (3) : 587613. doi: 10.3934/krm.2015.8.587 
[5] 
Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, nonnegativity and stability of a new Milstein scheme with applications to finance. Discrete & Continuous Dynamical Systems  B, 2013, 18 (8) : 20832100. doi: 10.3934/dcdsb.2013.18.2083 
[6] 
Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems  A, 2009, 23 (1&2) : 127. doi: 10.3934/dcds.2009.23.1 
[7] 
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$rule. Inverse Problems & Imaging, 2012, 6 (1) : 133146. doi: 10.3934/ipi.2012.6.133 
[8] 
Benoît Merlet, Morgan Pierre. Convergence to equilibrium for the backward Euler scheme and applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 685702. doi: 10.3934/cpaa.2010.9.685 
[9] 
WeiZhe Gu, LiYong Lu. The linear convergence of a derivativefree descent method for nonlinear complementarity problems. Journal of Industrial & Management Optimization, 2017, 13 (2) : 531548. doi: 10.3934/jimo.2016030 
[10] 
Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems  B, 2015, 20 (1) : 2338. doi: 10.3934/dcdsb.2015.20.23 
[11] 
Xinfu Chen, Bei Hu, Jin Liang, Yajing Zhang. Convergence rate of free boundary of numerical scheme for American option. Discrete & Continuous Dynamical Systems  B, 2016, 21 (5) : 14351444. doi: 10.3934/dcdsb.2016004 
[12] 
Tadahisa Funaki, Yueyuan Gao, Danielle Hilhorst. Convergence of a finite volume scheme for a stochastic conservation law involving a $Q$brownian motion. Discrete & Continuous Dynamical Systems  B, 2018, 23 (4) : 14591502. doi: 10.3934/dcdsb.2018159 
[13] 
Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous secondorder gradientlike systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 23932416. doi: 10.3934/cpaa.2012.11.2393 
[14] 
Arnaud Debussche, Jacques Printems. Convergence of a semidiscrete scheme for the stochastic Kortewegde Vries equation. Discrete & Continuous Dynamical Systems  B, 2006, 6 (4) : 761781. doi: 10.3934/dcdsb.2006.6.761 
[15] 
Mostafa Bendahmane, Mauricio Sepúlveda. Convergence of a finite volume scheme for nonlocal reactiondiffusion systems modelling an epidemic disease. Discrete & Continuous Dynamical Systems  B, 2009, 11 (4) : 823853. doi: 10.3934/dcdsb.2009.11.823 
[16] 
Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving nonlinear aggregationbreakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713737. doi: 10.3934/krm.2014.7.713 
[17] 
Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary NavierStokes equations. Discrete & Continuous Dynamical Systems  B, 2017, 22 (9) : 34213438. doi: 10.3934/dcdsb.2017173 
[18] 
Chun Wang, TianZhou Xu. Stability of the nonlinear fractional differential equations with the rightsided RiemannLiouville fractional derivative. Discrete & Continuous Dynamical Systems  S, 2017, 10 (3) : 505521. doi: 10.3934/dcdss.2017025 
[19] 
Xiaoming Wang. On the coupled continuum pipe flow model (CCPF) for flows in karst aquifer. Discrete & Continuous Dynamical Systems  B, 2010, 13 (2) : 489501. doi: 10.3934/dcdsb.2010.13.489 
[20] 
Qingguang Guan, Max Gunzburger. Stability and convergence of timestepping methods for a nonlocal model for diffusion. Discrete & Continuous Dynamical Systems  B, 2015, 20 (5) : 13151335. doi: 10.3934/dcdsb.2015.20.1315 
2017 Impact Factor: 0.972
Tools
Article outline
[Back to Top]