AM  Vol.4 No.10 , October 2013
Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations
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.
References
[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.

 
 
Top