Skip to content
BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access September 25, 2015

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

  • Béla J. Szekeres and Ferenc Izsák
From the journal Open Mathematics


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.


[1] Treumann, R. A., Theory of super-diffusion for the magnetopause, Geophys. Res. Lett., 1997, 24, 1727–1730 10.1029/97GL01760Search in Google Scholar

[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 10.1038/nature06199Search in Google Scholar PubMed

[3] Benson, D. A., Wheatcraft, S. W., Meerschaert, M. M., Application of a fractional advection-dispersion equation, Water Resour. Res., 2000, 36, 1403–1412 10.1029/2000WR900031Search in Google Scholar

[4] Podlubny, I., Fractional differential equations, Academic Press Inc., San Diego, CA, 1999 Search in Google Scholar

[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 10.1142/S0218202512500546Search in Google Scholar

[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 10.1137/110833294Search in Google Scholar

[7] Meerschaert, M. M., Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 2004, 172, 65–77 10.1016/ in Google Scholar

[8] Deng, W. H., Chen, M., Efficient numerical algorithms for three-dimensional fractional partial diffusion equations, J. Comp. Math., 2014, 32, 371–391 10.4208/jcm.1401-m3893Search in Google Scholar

[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 10.1016/ in Google Scholar

[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 10.1016/ in Google Scholar

[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 10.1090/S0025-5718-2015-02917-2Search in Google Scholar

[12] Zhou, H., Tian, W., Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput., 2013, 56, 45–66 10.1007/s10915-012-9661-0Search in Google Scholar

[13] Atangana, A., On the stability and convergence of the time-fractional variable order telegraph equation, J. Comput. Phys., 2015, 293, 104–114 10.1016/ in Google Scholar

[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 10.1007/s11075-015-9990-9Search in Google Scholar

[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 10.1007/s10208-014-9208-xSearch in Google Scholar

[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 10.1007/s11425-013-4716-8Search in Google Scholar

[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 10.2478/s11534-013-0264-7Search in Google Scholar

[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 10.1016/ in Google Scholar

[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 Search in Google Scholar

[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 10.1016/S0034-4877(14)60015-XSearch in Google Scholar

[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 10.1155/2014/914725Search in Google Scholar

[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 10.1155/2014/365981Search in Google Scholar

[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 Search in Google Scholar

[24] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and applications of fractional differential equations, Elsevier Science B.V., Amsterdam, 2006 Search in Google Scholar

[25] Miller, K. S., Ross, B., An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons Inc., New York, 1993 Search in Google Scholar

[26] Samko, S. G., Kilbas, A. A., Marichev, O. I., Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993 Search in Google Scholar

[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 Search in Google Scholar

[28] Hilfer, R., Threefold Introduction to Fractional Derivatives, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2008 Search in Google Scholar

[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 10.1016/j.apm.2009.04.006Search in Google Scholar

[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 10.1093/imamat/hxn033Search in Google Scholar

[31] Adams, R. A., Fournier, J. J. F., Sobolev spaces, Academic Press, Amsterdam, 2003 Search in Google Scholar

[32] Gradshteyn, I. S., Ryzhik, I. M., Table of integrals, series, and products, 6th ed., Academic Press Inc., San Diego, CA, 2000 Search in Google Scholar

[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 10.1137/12086491XSearch in Google Scholar

[34] Bonito, A., Pasciak, J. E., Numerical Approximation of Fractional Powers of Elliptic Operators, Math. Comp., 2015, 84, 2083–2110 10.1090/S0025-5718-2015-02937-8Search in Google Scholar

[35] Szymczak, P., Ladd, A. J. C., Boundary conditions for stochastic solutions of the convection-diffusion equation, Phys. Rev. E, 2003, 68, 12 10.1103/PhysRevE.68.036704Search in Google Scholar PubMed

[36] Baeumer, B., Kovács, M., Meerschaert, M. M., Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl., 2008, 55, 2212–2226 10.1016/j.camwa.2007.11.012Search in Google Scholar

[37] Evans, L. C., Partial differential equations, American Mathematical Society, Providence, RI, 1998 Search in Google Scholar

Received: 2015-3-5
Accepted: 2015-5-15
Published Online: 2015-9-25

©2015 Béla J. Szekeres and Ferenc Izsák

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Downloaded on 28.2.2024 from
Scroll to top button