Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
2015 | 13 | 1 |
Tytuł artykułu

A finite difference method for fractional diffusion equations with Neumann boundary conditions

Treść / Zawartość
Warianty tytułu
Języki publikacji
A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The wellposedness of the obtained initial value problem is proved and it is pointed out that each extension is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald–Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved.
Opis fizyczny
  • Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest,
    Pázmány P. s. 1/C, Hungary
  • Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, H-1117, Budapest,
    Pázmány P. s. 1/C, Hungary
  • MTA-ELTE NumNet Research Group, Eötvös Loránd University, 1117 Budapest,
    Pázmány P. stny. 1C, Hungary
  • [1] Treumann, R. A., Theory of super-diffusion for the magnetopause, Geophys. Res. Lett., 1997, 24, 1727–1730 [Crossref]
  • [2] Edwards, A. M., Phillips, R. A., Watkins, N. W., Freeman, M. P., Murphy, E. J., Afanasyev, V., et al., Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer, 2007, Nature, 449, 1044–1048 [WoS]
  • [3] Benson, D. A., Wheatcraft, S. W., Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res., 2000, 36, 1403–1412 [Crossref]
  • [4] Podlubny, I., Fractional differential equations, Academic Press Inc., San Diego, CA, 1999
  • [5] Du, Q., Gunzburger, M., Lehoucq, R. B., Zhou, K., A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Mod. Meth. Appl. Sci., 2013, 23, 493–540 [Crossref]
  • [6] Du, Q., Gunzburger, M., Lehoucq, R. B., Zhou, K., Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints, SIAM Rev., 2012, 54, 667–696 [Crossref][WoS]
  • [7] Meerschaert, M. M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 2004, 172, 65–77
  • [8] Deng, W. H., Chen, M., Efficient numerical algorithms for three-dimensional fractional partial diffusion equations, J. Comp. Math., 2014, 32, 371–391 [Crossref]
  • [9] Tadjeran, C., Meerschaert, M. M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 2007, 220, 813–823
  • [10] Tadjeran, C., Meerschaert, M. M., Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 2006, 213, 205–213
  • [11] Tian, W., Zhou, H., Deng, W., A class of second order difference approximations for solving space fractional diffusion equations, Math. Comp., 2015, 84, 1703–1727 [Crossref]
  • [12] Zhou, H., Tian, W., Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput., 2013, 56, 45–66 [Crossref]
  • [13] Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 2015, 293, 104–114 [WoS]
  • [14] Bhrawy, A. H., Zaky, M.A., Van Gorder, R.A., A space-time Legendre spectral tau method for the two-sided space-time Caputo fractional diffusion-wave equation, Numer. Algorithms, 2015 (in press), DOI: 10.1007/s11075-015-9990-9 [Crossref]
  • [15] Nochetto, R., Otárola, E., Salgado, A., A PDE Approach to Fractional Diffusion in General Domains: A Priori Error Analysis, Found. Comput. Math., 2014, 1–59
  • [16] Huang, J., Nie, N., Tang, Y., A second order finite difference-spectral method for space fractional diffusion equations, Sci. Chin. Math., 2014, 57, 1303–1317 [Crossref]
  • [17] Doha, E., Bhrawy, A., Ezz-Eldien, S., Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method, Cent. Eur. J. of Phys., 2013, 11, 1494–1503 [WoS]
  • [18] Bhrawy, A. H., Zaky, M. A., A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations, J. Comput. Phys., 2015, 281, 876–895
  • [19] Bhrawy, A. H., Zaky, M.A., Machado, J. T., Efficient Legendre spectral tau algorithm for solving two-sided space-time Caputo fractional advection-dispersion equation, J. Vib. Control, 2015 (in press), DOI: 10.1177/107754631456683 [Crossref]
  • [20] Bhrawy, A. H., Baleanu, D., A spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection diffusion equations with variable coefficients, Rep. Math. Phys., 2013, 72, 219–233 [WoS][Crossref]
  • [21] Xu, S., Ling, X., Cattani, C., Xie, G., Yang, X., Zhao, Y., Local fractional Laplace variational iteration method for nonhomogeneous heat equations arising in fractal heat flow, Math. Probl. Eng., 2014, Art. ID 914725 [WoS]
  • [22] Yang, A., Li, J., Srivastava, H. M., Xie, G., Yang, X., Local fractional Laplace variational iteration method for solving linear partial differential equations with local fractional derivative, Discrete Dyn. Nat. Soc., 2014, Art. ID 365981 [WoS]
  • [23] Ilic, M., Liu, F., Turner, I., Anh, V., Numerical approximation of a fractional-in-space diffusion equation. I, Fract. Calc. Appl. Anal., 2005, 8, 323–341
  • [24] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006
  • [25] Miller, K. S., Ross, B., An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons Inc., New York, 1993
  • [26] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993
  • [27] Atangana, A., Secer, A., A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstr. Appl. Anal., 2013, DOI:10.1155/2013/27968 [Crossref]
  • [28] Hilfer, R., Threefold Introduction to Fractional Derivatives, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2008
  • [29] Yang, Q., Liu, F., Turner, I., Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model., 2010, 34, 200–218 [WoS][Crossref]
  • [30] Shen, S., Liu, F., Anh, V., Turner, I., The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation, IMA J. Appl. Math., 2008, 73, 850–872 [WoS]
  • [31] Adams, R. A., Fournier, J. J. F., Sobolev spaces, Academic Press, Amsterdam, 2003
  • [32] Gradshteyn, I. S., Ryzhik, I. M., Table of integrals, series, and products, 6th ed., Academic Press Inc., San Diego, CA, 2000
  • [33] Wang, H. and Basu, T. S., A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 2012, 34, A2444–A2458 [WoS]
  • [34] Bonito, A., Pasciak, J. E., Numerical Approximation of Fractional Powers of Elliptic Operators, Math. Comp., 2015, 84, 2083–2110 [Crossref]
  • [35] Szymczak, P., Ladd, A. J. C., Boundary conditions for stochastic solutions of the convection-diffusion equation, Phys. Rev. E, 2003, 68, 12
  • [36] Baeumer, B., Kovács, M., Meerschaert, M. M., Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl., 2008, 55, 2212–2226 [Crossref]
  • [37] Evans, L. C., Partial differential equations, American Mathematical Society, Providence, RI, 1998
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.