Abstract
In this paper, we consider the two-dimensional non-linear fractional reaction-subdiffusion equation. A novel compact numerical method which is second-order temporal accuracy and fourth-order spatial accuracy is derived. The stability and convergence of the compact numerical method have been discussed rigorously by means of the Fourier method. Finally, numerical examples are presented to show the effectiveness and the high-order accuracy of the compact numerical method.
Similar content being viewed by others
References
Abbaszadeh, M., Mohebbi, A.: A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term. Comput. Math. Appl. 66 (8), 1345–1359 (2013)
Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus. World Scientific, Singapore (2012)
Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34 (4), A2145–A2172 (2012)
Chen, C.M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198 (2), 754–769 (2008)
Chen, C.M., Liu, F., Burrage, K.: Numerical analysis for a variable-order nonlinear cable equation. J. Comput. Appl. Math. 236 (2), 209–224 (2011)
Chen, C.M., Liu, F., Turner, I., Anh, V., Chen, Y.: Numerical approximation for a variable-order non-linear fractional reaction-subdiffusion equation. Numer. Algorithms 63, 265–290 (2013)
Chen, C.M., Liu, F., Turner, I., Anh, V.: Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algorithms 54 (1), 1–21 (2010)
Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81 (277), 345–366 (2011)
Chen, S., Liu, F.: ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. J. Appl. Math. Comput. 26 (1–2), 295–311 (2008)
Cui, M.R.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228 (20), 7792–7804 (2009)
Cui, M.R.: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algorithms 62 (3), 383–409 (2013)
Deng, W.: Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227 (2), 1510–1522 (2007)
Deng, W., Li, C.: Finite difference methods and their physical constraints for the fractional klein-kramers equation. Numer. Methods Partial. Differ. Equ. 27 (6), 1561–1583 (2011)
Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265 (2), 229–248 (2002)
Diethelm, K.: Fractional Differential Equations, Theory and Numerical Treatment, vol. 93. Technical University of Braunschweig (2003)
Diethelm, K.: The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)
Giona, M., Roman, H.E.: Fractional diffusion equation for transport phenomena in random media. Phys. A 185 (1), 87–97 (1992)
Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1 (2), 167–191 (1998)
Jiang, X.Y., Qi, H.T.: Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative. J. Phys. A 45 (48), 485101(10pp) (2012)
Jiang, X.Y., Xu, M.Y., Qi, H.T.: The fractional diffusion model with an absorption term and modified Fick’s law for non-local transport processes. Nonlinear Anal. 11 (1), 262–269 (2010)
Jiang, X.Y., Chen, S.Z.: Analytical and numerical solutions of time fractional anomalous thermal diffusion equation in composite medium. ZAMM J. Appl. Math. Mech. / Z. Angew. Math. Mech. 1–9 (2013)
Li, C.P., Zhao, Z., Chen, Y.Q.: Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62 (3), 855–875 (2011)
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 (1), 12–20 (2007)
Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231 (1), 160–176 (2009)
Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64 (10), 2990–3007 (2012)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (1), 65–77 (2004)
Meerschaert, M.M., Scheffler, H.P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211 (1), 249–261 (2006)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (1), 1–77 (2000)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220 (2), 813–823 (2007)
Xu, H., Liao, S.J., You, X.C.: Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14 (4), 1152–1156 (2009)
Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50 (3), 1535–1555 (2012)
Zeng, F., Li, C., Liu, F.: High-order explicit-implicit numerical methods for nonlinear anomalous diffusion equations. Eur. Phys. J. 222 (8), 1885–1900 (2013)
Zeng, F., Li, C., Liu, F., Turner, I.: The use of finite difference/element approximations for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35 (6), 2976–3000 (2013)
Zhuang, P., Liu, F.: Finite difference approximation for two-dimensional time fractional diffusion equation. J. Algorithms Comput. Technol. 1 (1), 1–15 (2007)
Zhuang, P., Liu, F., Anh, V., Turner, I.: Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74 (5), 645–667 (2009)
Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46 (2), 1079–1095 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Yu, B., Jiang, X. & Xu, H. A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation. Numer Algor 68, 923–950 (2015). https://doi.org/10.1007/s11075-014-9877-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-014-9877-1