Modelling the movement of groundwater pollution with variable order derivative
-
1944
Downloads
-
3644
Views
Authors
S. N. Kameni
- African Institute for Mathematical Sciences (AIMS), P. O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon.
J. D. Djida
- African Institute for Mathematical Sciences (AIMS), P. O. Box 608, Limbe Crystal Gardens, South West Region, Cameroon.
A. Atangana
- Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9301, Bloemfontein, South Africa.
Abstract
In this paper, a new concept of variable differentiation is used to revisit the model of groundwater pollution. The new variable order derivation has a non-singular kernel and can be used for analytical and numerical purposes. The novel model is solved via Fourier transform method. We solve numerically the new equation using the implicit finite difference scheme and study the stability and convergence of that scheme.
Share and Cite
ISRP Style
S. N. Kameni, J. D. Djida, A. Atangana, Modelling the movement of groundwater pollution with variable order derivative, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 10, 5422--5432
AMA Style
Kameni S. N., Djida J. D., Atangana A., Modelling the movement of groundwater pollution with variable order derivative. J. Nonlinear Sci. Appl. (2017); 10(10):5422--5432
Chicago/Turabian Style
Kameni, S. N., Djida, J. D., Atangana, A.. "Modelling the movement of groundwater pollution with variable order derivative." Journal of Nonlinear Sciences and Applications, 10, no. 10 (2017): 5422--5432
Keywords
- New variable order derivative
- advection-dispersion equation with variable order
- stability
- convergence
- finite difference scheme
MSC
References
-
[1]
L. M. Abriola, , Modeling contaminant transport in the subsurface: An interdisciplinary challenge, Rev. Geophys., 25 (1987), 125–134.
-
[2]
A. N. Angelakis, T. N. Kadir, D. E. Rolston, Solutions for transport of two sorbed solutes with differing dispersion coefficients in soil , Soil Sci. Soc. Am. J., 51 (1987), 1428–1434.
-
[3]
G. B. Arfken, H. J. Weber , Mathematical Methods for Physicists, Am. J. Phys., 67 (1999), 165–169.
-
[4]
A. Atangana, A Derivative with variable order with no singular kernel, J. Comput. Phys. , preprint (2016)
-
[5]
A. Atangana, A. Kilicman , Analytical solutions of the space-time fractional derivative of advection dispersion equation, Math. Probl. Engin., 2013 (2013), 9 pages.
-
[6]
D. A. Benson, S. W. Wheatcraft, M. M. Meerschaert , Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403–1412.
-
[7]
B. R. Bicknell, J. C. Imhoff, J. Kittle, L. John, J. Donigian, S. Anthony, R. C. Johanson, Hydrological simulation program–Fortran, User’s manual for version 11, US EPA (1996)
-
[8]
D. K. Jaiswal, A. Kumar, N. Kumar, M. K. Singh, Solute transport along temporally and spatially dependent flows through horizontal semi-infinite media: dispersion proportional to square of velocity, J. Hydrol. Eng., 16 (2011), 228–238.
-
[9]
D. K. Jaiswal, A. Kumar, N. Kumar, R. Yadav, Analytical solutions for temporally and spatially dependent solute dispersion of pulse type input concentration in one-dimensional semi-infinite media, J. Hydro. Environ. Res., 2 (2009), 254–263.
-
[10]
K. R. Lassey , Unidimensional solute transport incorporating equilibrium and ratelimited isotherms with firstorder loss: 1. Model conceptualizations and analytic solutions, Water Resour. Res., 24 (1988), 343–350.
-
[11]
R. Leonard, W. Knisel, F. Davis, A. Johnson , Validating GLEAMS with field data for fenamiphos and its metabolites, J. Irrig. Drain. Eng., 116 (1990), 24–35.
-
[12]
M. M. Meerschaert, C. Tadjeran , Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65–77.
-
[13]
T. G. Naymik, R. A. Freeze , Mathematical modeling of solute transport in the subsurface, Crit. Rev. Environ. Sci. Technol., 17 (1987), 229–251.
-
[14]
I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic press (1998)
-
[15]
B. K. Rao, D. L. Hathaway , A Three-Dimensional Mixing Cell Solute Transport Model and Its Application, Ground Water, 27 (1989), 509–516.
-
[16]
L. L. Shoemaker, W. L. Magette, A. Shirmohammadi, Modeling management practice effects on pesticide movement to ground water, Groundwater Monit. Remediat., 10 (1990), 109–115.
-
[17]
C. S. Slichter, Field measurements of the rate of movement of underground waters, Govt. Print. Off., Washington (1905)
-
[18]
W. J. Weber, C. T. Miller , Modeling the sorption of hydrophobic contaminants by aquifer materialsI. Rates and equilibria, Water Res., 22 (1988), 457–464.
-
[19]
L.Wen, X. L. Tang, Numerical Solving Two-dimensional Variable-order Fractional Advection-dispersion Equation, WSEAS Trans. Math., 2013 (2013), 7 pages.
-
[20]
R. Yadav, D. K. Jaiswal , Two-dimensional solute transport for periodic flow in isotropic porous media: an analytical solution, Hydrol. Process., 26 (2012), 3425–3433.