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Numerical Method Based on Galerkin Approximation for the Fractional Advection-Dispersion Equation

International Journal of Applied and Computational Mathematics volume 3, pages 2171–2187 (2017)Cite this article

Abstract

In this paper, we present a numerical method for solving homogeneous as well as non-homogeneous fractional advection-dispersion equation (FADE). The numerical method is based on the Galerkin approximation. The Galerkin approximation converts the FADE into a system of fractional ordinary differential equations whose solutions is discussed. Convergence of the proposed method is shown. Numerical examples are given and numerical results are compared with exact solution to show the effectiveness and accuracy of the proposed method.

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References

  1. Baeumer, B., Meerschaert, M.M., Benson, D.A., Wheatcraft, S.W.: Subordinate advection-dispersion equation for contaminant transport. Water Resour. Res. 37, 1543–1550 (2001)

    Article  Google Scholar 

  2. Bonilla, B., Rivero, M., Trujillo, J.J.: On system of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)

    MathSciNet  MATH  Google Scholar 

  3. Dehghan, M.: Numerical solution of three-dimensional advection-diffusion equation. Appl. Math. Comput. 150, 5–19 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Dehghan, M.: Fully implicit finite difference methods for two-dimensional diffusion with non-local boundary condition. J. Comput. Appl. Math. 106, 255–269 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  5. Dehghan, M.: Fully explicit finite difference methods for two-dimensional diffusion with an integer condition. Nonlinear Anal. Theory Methods Appl. 48, 637–650 (2002)

    Article  MATH  Google Scholar 

  6. Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Yu.: Algorithms for fractional calculus : a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  7. Gorenflo, R., Mainardi, F., Scalas, E., Roberto, M.: Fractional calculus and continuous time finance, III. The diffusion limit, in: Math. Finance, Konstanz, 2000, in: Trends Math, pp. 171–180. Birkhauser, Basel (2001)

  8. Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Book  MATH  Google Scholar 

  9. Huang, F., Liu, F.: The fundamental solution of the space-time fractional advection-dispersion. J. Appl. Math. Comput. 18, 339–350 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  10. Joseph, K.T., Sahoo, M.R.: Vanishing viscosity approach to a system of conservation laws admitting \(\delta ^{{\prime }{\prime }}\)–waves. Commun. pure. Appl. Anal. 12, 2091–2118 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Lin, Y., Jiang, W.: Approximate solution of the fractional advection-dispersion equation. Comp. Phys. Commun. 181, 557–561 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  12. Liu, F., Anh, V.V., Turner, I., Zhuang, P.: Time fractional advection-dispersion equation. J. Appl. Math. Comput. 13, 233–245 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  13. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection diffusion equation. Appl. Math. Comput. 191, 12–20 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Meerschaaert, M., Benson, D., Scheffler, H.P., Baeumer, B.: Stochastic solution of space-time fractional diffusion equation. Phys. Rev. E 65, 1103–1106 (2002)

    MathSciNet  Article  Google Scholar 

  15. Metzler, R., Klafter, J.: The restaurant at the end of the random walk: recent developments in the description of the anomalous transport by fractional dynamics. J. Phys. A 37, R161–R208 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  16. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamic approach. Phys. Rep. 339, 1–77 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  17. Mechado, J.T.: Discrete time fractional-order controllers. Fract. Calc. Appl. Anal. 4, 47–66 (2001)

    MathSciNet  Google Scholar 

  18. Miller, K., Ross, B.: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  19. Pandey, R.K., Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space-time fractional advection-dispersion equation. Comp. Phys. Commun. 182, 1134–1144 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  20. Pandey, R.K., Singh, O.P., Baranwal, V.K., Tripathi, M.P.: An analytic solution for the space-time fractional advection-dispersion equation using optimal homotopy asymptotic method. Comp. Phys. Commun. 183, 2098–2106 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  21. Panov, E.Y., Shelkovich, V.M.: \(\delta ^{{\prime }}\)-shock waves as a new type of solutions to systems of conservation laws. Electron. J. Diff. Equ. 28(1), 1–12 (2015)

    Google Scholar 

  22. Riemus, P., Pohll, G., Mihevc, T., Chapman, J., Haga, M., Lyles, B., Kosinski, S., Niswonger, R., Sanders, P.: Testing and parameterizing a conceptual for solute transport in fractured granite using multiple tracers in a forced-gradient test. Water Resour. Res. 39, 1356–1370 (2003)

    Google Scholar 

  23. Roberto, M., Scalas, E., Mainardi, F.: Waiting-times and returns in high frequency financial data: an empirical study. Phys, A 314, 749–755 (2002)

    Article  MATH  Google Scholar 

  24. Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distributions in financial markets. Eur. Phys. J. B. 27, 273–275 (2002)

    MathSciNet  Google Scholar 

  25. Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B., Wheatcraft, S.W.: Eulerian derivation of the fractional-dispersion equation. J. Contam. Hydrol. 48, 69–88 (2001)

    Article  Google Scholar 

  26. Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Multiscaling fractional advection-dispersion equation and their solutions. Water Resour. Res. 39, 1022–1032 (2003)

    Google Scholar 

  27. El-Sayed, A.M.A., Behiry, S.H., Raslan, W.E.: Adomian’s decomposition method for solving an intermediate advection-dispersion equation. Comput. Math. Appl. 59, 1759–1765 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  28. Singh, H., Sahoo, M.R., Singh, O.P.: Weak asymptotic solution to a non-strictly system of conservation laws. Electron. J. Diff. Equ. 1, 1–11 (2015)

    MathSciNet  MATH  Google Scholar 

  29. West, B., Bologna, M., Grigolini, P.: Physics of Fractal Operators. Springer, New York (2003)

    Book  Google Scholar 

  30. Zaslavsky, G.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

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Acknowledgments

The author is very grateful to the referees for their constructive comments and suggestions for the improvement of the paper. This paper is dedicated in the memory of Late Professor Om Prakash Singh who is one of the co-author of this article.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi, 221005, India

    Harendra Singh & Om Prakash Singh

  2. School of Mathematical Sciences, National Institute of Science Education and Research (NISER), Khurda, Odisha, 752050, India

    Manas Ranjan Sahoo

Authors
  1. Harendra Singh
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  2. Manas Ranjan Sahoo
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  3. Om Prakash Singh
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Correspondence to Harendra Singh.

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Singh, H., Sahoo, M.R. & Singh, O.P. Numerical Method Based on Galerkin Approximation for the Fractional Advection-Dispersion Equation. Int. J. Appl. Comput. Math 3, 2171–2187 (2017). https://doi.org/10.1007/s40819-016-0233-0

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  • DOI: https://doi.org/10.1007/s40819-016-0233-0

Keywords

  • Fractional order advection-dispersion equation
  • Galerkin approximation
  • System of fractional order ODE
  • Convergence analysis

Mathematics Subject Classification

  • 35Q99
  • 65N99
  • 65D99