Numerical approximation for a variable-order nonlinear reaction-subdiffusion equation

Numerical Algorithms

Volumn 63, Issue 2, 2013, Pages 265-290

  Chen, Chang Ming ^a   Liu, F ^b   Turner, I ^b   Anh, V ^b   Chen, Y ^a  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Convergence and stability; Improved numerical approximation; Nonlinear reaction subdiffusion equation; Variable order Riemann Liouville partial derivative

Indexed keywords

EID: 84878321898     PISSN: 10171398     EISSN: 15729265     Source Type: Journal    
DOI: 10.1007/s11075-012-9622-6     Document Type: Article

References (44)

1. Agrawal, O. P.: Solution for a fractional diffusion-wave equation defined in a boundary domain. Nonlinear Dyn. 29, 145-155 (2002).
2. Baeumer, B., Koávcs, M., Meerschaert, M.: Numerical solutions for fractional reaction-diffusion equations. Comput. Math. Appl. 55, 2212-2226 (2008).
3. Chen, C.-M., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198, 754-769 (2008).
4. Chen, C.-M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740-1760 (2010).
5. Chen, C.-M., Liu, F., Anh, V., Turner, I.: Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term. Appl. Math. Comput. 217, 5729-5742 (2011).
6. Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker-Planck equation. Appl. Math. Model. 33, 256-273 (2009).
7. Chiu, J. W., Chiam, K.-H.: Monte Carlo simulation and linear stability analysis of Turing pattern formation in reaction-subdiffusion systems. Phys. Rev. E 78, 056708 (2008).
8. Coimbra, C. F. M.: Mechanics with variable-order differential operators. Ann. Phys. 12, 692-703 (2003).
9. Evans, K. P., Jacob, N.: Feller semigroups obtained by variable-order subordination. Rev. Mat. Complut. 20(2), 293-307 (2007).
10. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Carpinteri, A., Maainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics, pp. 223-276. Springer, New York (1997).
11. Jacob, N., Leopold, H.: Pseudo differential operators with variable order of differentiation generating Feller semigroup. Integr. Equ. Oper. Theory 17, 544-553 (1993).
12. Kikuchi, K., Negoro, A.: On Markov processes generated by pseudodifferentail operator of variable order. Osaka J. Math. 34, 319-335 (1997).
13. Langlands, T. A. M., Henry, B. I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comp. Physiol. 205, 719-736 (2005).
14. Leopold, H. G.: Embedding of function spaces of variable order of differentiation. Czechoslov. Math. J. 49, 633-644 (1999).
15. Lin, R., Liu, F., Anh, V., Turner, I.: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput. 212, 435-445 (2009).
16. Lin, Y., Xu, C.: Finite difference/spectral approximation for the time-fractional diffusion equation. J. Comp. Physiol. 225, 1533-1552 (2007).
17. Liu, F., Zhang, P., Anh, V., Burrage, K.: Stability and convergence of the difference methods for the space-time feactional advection-diffusion equation. Appl. Math. Comput. 191, 12-20 (2007).
18. 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, 160-176 (2009).
19. Liu, Q., Liu, F., Turner, I., Anh, V.: Approximation of the Levy-Feller advection-dispersion process by random walk and finite difference method. J. Comp. Physiol. 222, 57-70 (2007).
20. Lorenzo, C. F., Hartley, T. T.: Initialization, conceptualization and application in the generalized fractional calculus. NASA/TP-1998-208-208415 (1998).
21. Lorenzo, C. F., Hartley, T. T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57-98 (2002).
22. Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65-77 (2004).
23. Metzler, R., Klafter, J.: The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1-77 (2000).
24. Morton, K. W., Mayers, D. F.: Numerical Solution of Partial Differential Equations. Cambridge University Press, Cambridge (1994).
25. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999).
26. Ramirez, L. E. S., Coimbra, C. F. M.: Variable order constitutive relation for viscoelasticity. Ann. Phys. 16, 543-552 (2007).
27. Ramirez, L. E. S., Coimbra, C. F. M.: On the selection and meaning of variable order operators for dynamic modeling. Int. J. Differ. Equ. 2010, Article ID 846107 (2010). doi: 10. 1155/2010/846107.
28. Ruiz-Medina, M. D., Anh, V. V., Angulo, J. M.: Fractional generalized random fields of variable order. Stoch. Anal. Appl. 22, 775-799 (2004).
29. Sagués, F., Shkilev, V. P., Sokolov, I. M.: Reaction-subdiffusion equations for the A⇌B reaction. Phys. Rev. E 77, 032102 (2008).
30. Samko, S. G., Ross, B.: Integration and differentiation to a variable fractional order. Integr. Transf. Spec. F. 1, 277-300 (1993).
31. Schmidt-Martens, H. H., Froemberg, D., Sokolov, I. M.: Front propagation in a one-dimensional autocatalytic reaction-subdiffusion system. Phys. Rev. E 79, 041135 (2009).
32. 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. 73, 850-872 (2008).
33. Sokolov, I. M., Schmidt, M. G. W., Sagués, F.: Reaction-subdiffusion equations. Phys. Rev. E 73, 031102 (2006).
34. Smith, G. D.: Numerical Solution of Partial Differential Equations: Finite Differece Methods. Oxford Press, Toronto (1985).
35. Soon, C. M., Coimbra, C. F. M., Kobayashi, M. H.: Variable viscoelasticity operator. Ann. Phys. 14, 378-389 (2005).
36. Taddjeran, C., Meerschaert, M., Scheffler, H.: A second-order accuate numerical approximation for the fractional diffusion equation. J. Comp. Physiol. 213, 205-213 (2006).
37. Yadav, A., Horsthemke, W.: Kinetic equations for reaction-subdiffusion systems: Derivation and stability analysis. Phys. Rev. E 74, 066118 (2006).
38. Yadav, A., Milu, S. M., Horsthemke, W.: Turing instability in reaction-subdiffusion systems. Phys. Rev. E 78, 026116 (2008).
39. Yu, Q., Liu, F., Anh, V., Turner, I.: Solving linear and non-linear space-time fractional reaction-diffusion equations by the Adomian decomposition method. Int. J. Numer. Methods Eng. 74, 138-158 (2008).
40. Yuste, S. B., Acedo, L., Lindenberg, K.: Reaction front in an A+B →C reaction-subdiffusion process. Phys. Rev. E 69, 036126 (2004).
41. Yuste, S. B., Acedo, L.: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862-1874 (2005).
42. Yuste, S. B.: Weighted average finite difference methods for fractional diffusion equations. J. Comp. Physiol. 216, 264-274 (2006).
43. Zhang, H., Liu, F., Anh, V.: Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation. J. Comput. Appl. Math. 206, 1098-1115 (2007).
44. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760-1781 (2009).