Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations

Author(s)
E. A. Abdel-Rehim

Affiliation(s)

Department of Mathematics and Computer Science, Faculty of Science, Suez Canal University, Ismailia, Egypt.

Department of Mathematics and Computer Science, Faculty of Science, Suez Canal University, Ismailia, Egypt.

ABSTRACT

The space-time fractional
advection dispersion equations are linear partial pseudo-differential equations with spatial fractional derivatives in
time and in space and are used to model transport at the earth surface. The
time fractional order is denoted by *β*∈* * and is devoted to the space fractional order. The
time fractional advection dispersion equations describe particle motion with
memory in time. Space-fractional advection dispersion equations arise when
velocity variations are heavy-tailed
and describe particle motion that accounts for variation in the flow field over
entire system. In this
paper, I focus on finding the
precise explicit discrete approximate solutions to these models for some values
of with , while the Cauchy case as and the classical case as with are studied separately. I compare the
numerical results of these models for different values of and and for some other
related changes. The approximate solutions of these models are also discussed
as a random walk with or without a memory depending on the value of . Then I prove that the
discrete solution in the Fourierlaplace space of theses models converges in distribution
to the Fourier-Laplace transform of the corresponding fractional differential equations
for all the fractional values of and .

Cite this paper

Abdel-Rehim, E. (2013) Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations.*Applied Mathematics*, **4**, 1427-1440. doi: 10.4236/am.2013.410193.

Abdel-Rehim, E. (2013) Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations.

References

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[2] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, Boston, New York, London, Sydney, Tokyo, Toronto, 1999.

[3] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integrals and Derivatives (Theory and Applications),” Gordon and Breach, New York, London, and Paris, 1993.

[4] R. Gorenflo and F. Mainardi, “Fractional Calculus: Integral and Differential Equations of Fractional Order,” In: A. Carpinteri and F. Mainardi, Eds., Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 1997, pp. 223-276. http://www.fracalmo.org

[5] N. U. Prabhu, “Stochastic Processes (Basic Theory and Its Applications),” The Macmillan Company, New York, Collier-Macmillan Limited, London, 1965.

[6] E. A. Abdel-Rehim, “Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks,” Mensch&Buch Verlag, 2004. http://www.diss.fu-berlin.de/2004/168/index.html

[7] W. Feller, “On a Generalization of Marcel Riesz’ Potentials and the Semi-Groups Generated by Them,” In: Meddelanden Lunds Universitetes Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome Suppl. dédié a M. Riesz, Lund, 1952, pp. 73-81.

[8] M. M. Meerschaert and C. Tadjeran, “Finite Difference Approximations for Fractional Advection-Dispersion Flow Equation,” Journal of Computational and Applied Mathematics, Vol. 172, No. 1, 2004, pp. 65-77. http://dx.doi.org/10.1016/j.cam.2004.01.033

[9] D. A.Benson, S. W. Wheatcraft and M. M. Meerschaert, “Application of a Fractional Advection-Dispersion Equation,” Water Resource Research, Vol. 36, No. 6, 2000, pp. 1403-1412.

http://dx.doi.org/10.1029/2000WR900031

[10] D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, “Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests,” Transport in Porous Media, Vol. 42, No. 1-2, 2001, pp. 211-240. http://dx.doi.org/10.1023/A:1006733002131

[11] B. Baeumer, D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft, “Subordinated Advection-Dispersion Equation for Contaminant Transport,” Water Resource Research, Vol. 37, No. 6, 2001, pp. 1543-1550.

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[13] F. Huang and F. Liu, “The Fundamental Solution of the spaace-time Fractional advection-dispersion Equation,” Journal of Applied Mathematics and Computing Vol. 18, No. 1-2, 2005, pp. 339-350.

[14] Y.-S. Park and J.-J. Baik, “Analytical Solution of the Advection-Diffusion Equation for a Ground-Level Finite Area Source,” Atomspheric Environment, Vol. 42, No. 40, 2008, pp. 9603-9069.

http://dx.doi.org/10.1016/j.atmosenv.2008.09.019

[15] D. K. Jaiswal, A. Kumar and R. R. Yadav, “Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients,” Journal of Water Resource and Protection, Vol. 3, No. 1, 2011, pp. 76-84. http://dx.doi.org/10.4236/jwarp.2011.31009

[16] Y. Xia, J. C. Wu and L. Y. Zhou, “Numerical Solutions of Time-Space Fractional Advection-Dispersion Equations,” ICCES, Vol. 9, No. 2, 2009, pp. 117-126.

[17] Q. Liu, F. Liu, I. Turner and V. Anh, “Approximation of the Lévy-Feller Advection-Dispersion Process by Random Walk and Finite Difference Method,” Journal of Computational Physics, Vol. 222, No. 1, 2007, pp. 57-70. http://dx.doi.org/10.1016/j.jcp.2006.06.005

[18] F. Mainardi, Y. Luchko and G. Pagnini, “The Fundamental Solution of the Space-Time Fractional Diffusion Equation,” Fractional Calculus and Applied Analysis, Vol. 4, No. 2, 2001, pp. 153-192. www.fracalmo.org

[19] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, “Time-Fractional Diffusion: A Discrete Random Walk Approach,” Nonlinear Dynamics, Vol. 29, No. 1-4, 2002, pp. 129-143.

http://dx.doi.org/10.1023/A:1016547232119

[20] R. Gorenflo and E. A. Abdel-Rehim, “Discrete Models of Time-Fractional Diffusion in a Potential Well,” Fractional Calculus and Applied Analysis, Vol. 8, No. 2, 2005, pp. 173-200.

[21] R. Gorenflo and E. A. Abdel-Rehim, “From Power Laws to Fractional Diffusion: The Direct Way,” Vietnam Journal of Mathematics, Vol. 32, No. SI, 2004, pp. 65-75.

[22] R. Gorenflo and E. A. Abdel-Rehim, “Convergence of the Grünwald-Letnikov Scheme for Time-Fractional Diffusion,” Journal of Computational and Applied Mathematics, Vol. 205, No. 2, 2007, pp. 871-881. http://dx.doi.org/10.1016/j.cam.2005.12.043

[23] R. Gorenflo and E. A. Abdel-Rehim, “Simulation of Continuous Time Random Walk of the Space-Fractional Diffusion Equations,” Journal of Computational and Applied Mathematics, Vol. 222, No. 2, 2008, pp. 274-285. http://dx.doi.org/10.1016/j.cam.2007.10.052

[24] E. A. Abdel-Rehim, “From the Ehrenfest Model to TimeFractional Stochastic Processes,” Journal of Computational and Applied Mathematics, Vol. 233, No. 2, 2009, pp. 197-207.

http://dx.doi.org/10.1016/j.cam.2009.07.010

[25] A. I. Saichev and G. M. Zaslavsky, “Fractional Kinetic Equations: Solutions and Applications,” Chaos, Vol. 7, No. 4, 1997, pp. 753-764. http://dx.doi.org/10.1063/1.166272

[26] J. A. Goldstein, “Semigroups of Linear Operators and Applications,” Oxford University Press, Oxford and New York, 1985.

[27] N. Jacob, “Pseudo-Differential Operators and Markov Processes,” Akademie Verlag, Berlin, 1996.

[28] R. Metzler, J. Klafter and I. M. Sokolov, “Anomalous Transported in External Fields: Continuous Time Random Walks and Fractional Diffusion Equations Extended,” Physical Review E, Vol. 48, No. 2, 1998, pp. 1621-1633. http://dx.doi.org/10.1103/PhysRevE.58.1621

[29] W. Feller, “An Introduction to Probability Theory and Its Applications,” Vol. 2, Johon Wiley and Sons, New York, London, Sydney, Toronto, 1971.

[30] M. Kac, “Random Walk and the Theory of Brownian Motion,” The American Mathematical Monthly, Vol. 54, No. 7, 1947, pp. 369-391. http://dx.doi.org/10.2307/2304386

[31] K. B. Oldham and J. Spanier, “The Fractional Calculus,” Vol. 3 of Mathematics in Science and Engineering, Academic Press, New York, 1974.

[32] R. Gorenflo and F. Mainardi, “Random Walk Models for Space-Fractional Diffusion Processes,” Fractional Calculus and Applied Analysis, Vol. 1, No. 2, 1998, pp. 167190.

[33] R. Gorenflo and F. Mainardi, “Approximation of LévyFeller Diffusion by Random Walk,” Journal of Analysis and its Applications (ZAA) Vol. 18, No. 2, 1999, pp. 231246.

[1] K. S. Miller and B. Ross, “An Introduction to the Fractional Calculus and Fractional Differential Equations,” John Wily and Sons, INC., New York, Chichester, Brisbane, Toronto, Singapore, 1993.

[2] I. Podlubny, “Fractional Differential Equations,” Academic Press, San Diego, Boston, New York, London, Sydney, Tokyo, Toronto, 1999.

[3] S. G. Samko, A. A. Kilbas and O. I. Marichev, “Fractional Integrals and Derivatives (Theory and Applications),” Gordon and Breach, New York, London, and Paris, 1993.

[4] R. Gorenflo and F. Mainardi, “Fractional Calculus: Integral and Differential Equations of Fractional Order,” In: A. Carpinteri and F. Mainardi, Eds., Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 1997, pp. 223-276. http://www.fracalmo.org

[5] N. U. Prabhu, “Stochastic Processes (Basic Theory and Its Applications),” The Macmillan Company, New York, Collier-Macmillan Limited, London, 1965.

[6] E. A. Abdel-Rehim, “Modelling and Simulating of Classical and Non-Classical Diffusion Processes by Random Walks,” Mensch&Buch Verlag, 2004. http://www.diss.fu-berlin.de/2004/168/index.html

[7] W. Feller, “On a Generalization of Marcel Riesz’ Potentials and the Semi-Groups Generated by Them,” In: Meddelanden Lunds Universitetes Matematiska Seminarium (Comm. Sém. Mathém. Université de Lund), Tome Suppl. dédié a M. Riesz, Lund, 1952, pp. 73-81.

[8] M. M. Meerschaert and C. Tadjeran, “Finite Difference Approximations for Fractional Advection-Dispersion Flow Equation,” Journal of Computational and Applied Mathematics, Vol. 172, No. 1, 2004, pp. 65-77. http://dx.doi.org/10.1016/j.cam.2004.01.033

[9] D. A.Benson, S. W. Wheatcraft and M. M. Meerschaert, “Application of a Fractional Advection-Dispersion Equation,” Water Resource Research, Vol. 36, No. 6, 2000, pp. 1403-1412.

http://dx.doi.org/10.1029/2000WR900031

[10] D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, “Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests,” Transport in Porous Media, Vol. 42, No. 1-2, 2001, pp. 211-240. http://dx.doi.org/10.1023/A:1006733002131

[11] B. Baeumer, D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft, “Subordinated Advection-Dispersion Equation for Contaminant Transport,” Water Resource Research, Vol. 37, No. 6, 2001, pp. 1543-1550.

[12] R. Schumer, M. M. Meerschaert and B. Baeumer, “Fractional Advection-Dispersion Equations for Modeling Transport at the Earth Surface,” Journal of Geophysical research, Vol. 114, No. F4, 2009. http://dx.doi.org/10.1029/2008JF001246

[13] F. Huang and F. Liu, “The Fundamental Solution of the spaace-time Fractional advection-dispersion Equation,” Journal of Applied Mathematics and Computing Vol. 18, No. 1-2, 2005, pp. 339-350.

[14] Y.-S. Park and J.-J. Baik, “Analytical Solution of the Advection-Diffusion Equation for a Ground-Level Finite Area Source,” Atomspheric Environment, Vol. 42, No. 40, 2008, pp. 9603-9069.

http://dx.doi.org/10.1016/j.atmosenv.2008.09.019

[15] D. K. Jaiswal, A. Kumar and R. R. Yadav, “Analytical Solution to the One-Dimensional Advection-Diffusion Equation with Temporally Dependent Coefficients,” Journal of Water Resource and Protection, Vol. 3, No. 1, 2011, pp. 76-84. http://dx.doi.org/10.4236/jwarp.2011.31009

[16] Y. Xia, J. C. Wu and L. Y. Zhou, “Numerical Solutions of Time-Space Fractional Advection-Dispersion Equations,” ICCES, Vol. 9, No. 2, 2009, pp. 117-126.

[17] Q. Liu, F. Liu, I. Turner and V. Anh, “Approximation of the Lévy-Feller Advection-Dispersion Process by Random Walk and Finite Difference Method,” Journal of Computational Physics, Vol. 222, No. 1, 2007, pp. 57-70. http://dx.doi.org/10.1016/j.jcp.2006.06.005

[18] F. Mainardi, Y. Luchko and G. Pagnini, “The Fundamental Solution of the Space-Time Fractional Diffusion Equation,” Fractional Calculus and Applied Analysis, Vol. 4, No. 2, 2001, pp. 153-192. www.fracalmo.org

[19] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi, “Time-Fractional Diffusion: A Discrete Random Walk Approach,” Nonlinear Dynamics, Vol. 29, No. 1-4, 2002, pp. 129-143.

http://dx.doi.org/10.1023/A:1016547232119

[20] R. Gorenflo and E. A. Abdel-Rehim, “Discrete Models of Time-Fractional Diffusion in a Potential Well,” Fractional Calculus and Applied Analysis, Vol. 8, No. 2, 2005, pp. 173-200.

[21] R. Gorenflo and E. A. Abdel-Rehim, “From Power Laws to Fractional Diffusion: The Direct Way,” Vietnam Journal of Mathematics, Vol. 32, No. SI, 2004, pp. 65-75.

[22] R. Gorenflo and E. A. Abdel-Rehim, “Convergence of the Grünwald-Letnikov Scheme for Time-Fractional Diffusion,” Journal of Computational and Applied Mathematics, Vol. 205, No. 2, 2007, pp. 871-881. http://dx.doi.org/10.1016/j.cam.2005.12.043

[23] R. Gorenflo and E. A. Abdel-Rehim, “Simulation of Continuous Time Random Walk of the Space-Fractional Diffusion Equations,” Journal of Computational and Applied Mathematics, Vol. 222, No. 2, 2008, pp. 274-285. http://dx.doi.org/10.1016/j.cam.2007.10.052

[24] E. A. Abdel-Rehim, “From the Ehrenfest Model to TimeFractional Stochastic Processes,” Journal of Computational and Applied Mathematics, Vol. 233, No. 2, 2009, pp. 197-207.

http://dx.doi.org/10.1016/j.cam.2009.07.010

[25] A. I. Saichev and G. M. Zaslavsky, “Fractional Kinetic Equations: Solutions and Applications,” Chaos, Vol. 7, No. 4, 1997, pp. 753-764. http://dx.doi.org/10.1063/1.166272

[26] J. A. Goldstein, “Semigroups of Linear Operators and Applications,” Oxford University Press, Oxford and New York, 1985.

[27] N. Jacob, “Pseudo-Differential Operators and Markov Processes,” Akademie Verlag, Berlin, 1996.

[28] R. Metzler, J. Klafter and I. M. Sokolov, “Anomalous Transported in External Fields: Continuous Time Random Walks and Fractional Diffusion Equations Extended,” Physical Review E, Vol. 48, No. 2, 1998, pp. 1621-1633. http://dx.doi.org/10.1103/PhysRevE.58.1621

[29] W. Feller, “An Introduction to Probability Theory and Its Applications,” Vol. 2, Johon Wiley and Sons, New York, London, Sydney, Toronto, 1971.

[30] M. Kac, “Random Walk and the Theory of Brownian Motion,” The American Mathematical Monthly, Vol. 54, No. 7, 1947, pp. 369-391. http://dx.doi.org/10.2307/2304386

[31] K. B. Oldham and J. Spanier, “The Fractional Calculus,” Vol. 3 of Mathematics in Science and Engineering, Academic Press, New York, 1974.

[32] R. Gorenflo and F. Mainardi, “Random Walk Models for Space-Fractional Diffusion Processes,” Fractional Calculus and Applied Analysis, Vol. 1, No. 2, 1998, pp. 167190.

[33] R. Gorenflo and F. Mainardi, “Approximation of LévyFeller Diffusion by Random Walk,” Journal of Analysis and its Applications (ZAA) Vol. 18, No. 2, 1999, pp. 231246.