A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation

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• 作者：Bo Yu (1)
  Xiaoyun Jiang (1)
  Huanying Xu (2)
  
  1. School of Mathematics ; Shandong University ; Jinan ; 250100 ; China
  2. School of Mathematics and Statistics ; Shandong University ; Weihai ; Weihai ; 264209 ; China
  
• 关键词：Compact numerical method ; Two ; dimensional ; Non ; linear ; Stability ; Convergence ; Fourier method
• 刊名：Numerical Algorithms
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：68
• 期：4
• 页码：923-950
• 全文大小：573 KB
• 参考文献：1. Abbaszadeh, M, Mohebbi, A (2013) A fourth-order compact solution of the two-dimensional modified anomalous fractional sub-diffusion equation with a nonlinear source term. Comput. Math. Appl. 66: pp. 1345-1359 CrossRef
  2. Baleanu, D, Diethelm, K, Scalas, E, Trujillo, JJ (2012) Fractional Calculus. World Scientific, Singapore
  3. Burrage, K, Hale, N, Kay, D (2012) An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34: pp. A2145-A2172 CrossRef
  4. Chen, CM, Liu, F, Burrage, K (2008) Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198: pp. 754-769 CrossRef
  5. Chen, CM, Liu, F, Burrage, K (2011) Numerical analysis for a variable-order nonlinear cable equation. J. Comput. Appl. Math. 236: pp. 209-224 CrossRef
  6. Chen, CM, Liu, F, Turner, I, Anh, V, Chen, Y (2013) Numerical approximation for a variable-order non-linear fractional reaction-subdiffusion equation. Numer. Algorithms 63: pp. 265-290 CrossRef
  7. Chen, CM, Liu, F, Turner, I, Anh, V (2010) Numerical schemes and multivariate extrapolation of a two-dimensional anomalous sub-diffusion equation. Numer. Algorithms 54: pp. 1-21 CrossRef
  8. Chen, CM, Liu, F, Anh, V, Turner, I (2011) Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81: pp. 345-366 CrossRef
  9. Chen, S, Liu, F (2008) ADI-Euler and extrapolation methods for the two-dimensional fractional advection-dispersion equation. J. Appl. Math. Comput. 26: pp. 295-311 CrossRef
  10. Cui, MR (2009) Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228: pp. 7792-7804 CrossRef
  11. Cui, MR (2013) Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algorithms 62: pp. 383-409 CrossRef
  12. Deng, W (2007) Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227: pp. 1510-1522 CrossRef
  13. Deng, W, Li, C (2011) Finite difference methods and their physical constraints for the fractional klein-kramers equation. Numer. Methods Partial. Differ. Equ. 27: pp. 1561-1583 CrossRef
  14. Diethelm, K, Ford, NJ (2002) Analysis of fractional differential equations. J. Math. Anal. Appl. 265: pp. 229-248 CrossRef
  15. Diethelm, K.: Fractional Differential Equations, Theory and Numerical Treatment, vol. 93. Technical University of Braunschweig (2003)
  16. Diethelm, K (2010) The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin
  17. Giona, M, Roman, HE (1992) Fractional diffusion equation for transport phenomena in random media. Phys. A 185: pp. 87-97 CrossRef
  18. Gorenflo, R, Mainardi, F (1998) Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1: pp. 167-191
  19. Jiang, XY, Qi, HT (2012) Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative. J. Phys. A 45: pp. 485101(10pp) CrossRef
  20. Jiang, XY, Xu, MY, Qi, HT (2010) The fractional diffusion model with an absorption term and modified Fick鈥檚 law for non-local transport processes. Nonlinear Anal. 11: pp. 262-269 CrossRef
  21. 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鈥? (2013)
  22. Li, CP, Zhao, Z, Chen, YQ (2011) Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. Comput. Math. Appl. 62: pp. 855-875 CrossRef
  23. Liu, F, Zhuang, P, Anh, V, Turner, I, Burrage, K (2007) Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191: pp. 12-20 CrossRef
  24. Liu, F, Yang, C, Burrage, K (2009) Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231: pp. 160-176 CrossRef
  25. Liu, F, Zhuang, P, Burrage, K (2012) Numerical methods and analysis for a class of fractional advection-dispersion models. Comput. Math. Appl. 64: pp. 2990-3007 CrossRef
  26. Meerschaert, MM, Tadjeran, C (2004) Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172: pp. 65-77 CrossRef
  27. Meerschaert, MM, Scheffler, HP, Tadjeran, C (2006) Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211: pp. 249-261 CrossRef
  28. Metzler, R, Klafter, J (2000) The random walk鈥檚 guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339: pp. 1-77 CrossRef
  29. Podlubny, I (1999) Fractional Differential Equations. Academic Press, San Diego
  30. Tadjeran, C, Meerschaert, MM (2007) A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220: pp. 813-823 CrossRef
  31. Xu, H, Liao, SJ, You, XC (2009) Analysis of nonlinear fractional partial differential equations with the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 14: pp. 1152-1156 CrossRef
  32. Zhang, YN, Sun, ZZ, Zhao, X (2012) Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50: pp. 1535-1555 CrossRef
  33. Zeng, F, Li, C, Liu, F (2013) High-order explicit-implicit numerical methods for nonlinear anomalous diffusion equations. Eur. Phys. J. 222: pp. 1885-1900
  34. Zeng, F, Li, C, Liu, F, Turner, I (2013) The use of finite difference/element approximations for solving the time-fractional subdiffusion equation. SIAM J. Sci. Comput. 35: pp. 2976-3000 CrossRef
  35. Zhuang, P, Liu, F (2007) Finite difference approximation for two-dimensional time fractional diffusion equation. J. Algorithms Comput. Technol. 1: pp. 1-15 CrossRef
  36. Zhuang, P, Liu, F, Anh, V, Turner, I (2009) Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process. IMA J. Appl. Math. 74: pp. 645-667 CrossRef
  37. Zhuang, P, Liu, F, Anh, V, Turner, I (2008) New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation. SIAM J. Numer. Anal. 46: pp. 1079-1095 CrossRef
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

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.