Two new implicit numerical methods for the fractional cable equation

Journal of Computational and Nonlinear Dynamics

Volumn 6, Issue 1, 2011, Pages

  Liu, Fawang ^a   Yang, Qianqian ^a   Turner, Ian ^a  

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

CABLE EQUATION; CONVERGENCE ORDER; ENERGY METHOD; FRACTIONAL ORDER; FUNDAMENTAL EQUATIONS; IMPLICIT NUMERICAL METHOD; NEURONAL DYNAMICS; NUMERICAL RESULTS; PARTIAL DERIVATIVES; STABILITY AND CONVERGENCE; SUBDIFFUSION; TEMPORAL DERIVATIVES; TEMPORAL OPERATORS; TIME AND SPACE;

CABLES; MATHEMATICAL OPERATORS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 78649919287     PISSN: 15551415     EISSN: 15551423     Source Type: Journal    
DOI: 10.1115/1.4002269     Document Type: Article

References (40)

1. Yuste, S. B., and Lindenberg, K., 2001, "Subdiffusion-Limited A + A Reactions, " Phys. Rev. Lett., 87(11), p. 118301.
2. Barkai, E., Metzler, R., and Klafter, J., 2000, "From Continuous Time Random Walks to the Fractional Fokker-Planck Equation, " Phys. Rev. E, 61(1), pp. 132-138.
3. Metzler, R., and Klafter, J., 2000, "Boundary Value Problems for Fractional Diffusion Equations, " Physica A, 278 (1-2), pp. 107-125.
4. Metzler, R., and Klafter, J., 2000, "The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach, " Phys. Rep., 339(1), pp. 1-77.
5. Saichev, A. I., and Zaslavsky, G. M., 1997, "Fractional Kinetic Equations: Solutions and Applications, " Chaos, 7(4), pp. 753-764.
6. Zaslavsky, G. M., 2002, "Chaos, Fractional Kinetics, and Anomalous Transport, " Phys. Rep., 371(6), pp. 461-580.
7. Yuste, S. B., Acedo, L., and Lindenberg, K., 2004, "Reaction Front in an A +B-
8. Benson, D. A., Wheatcraft, S. W, and Meerschaert, M. M., 2000, "Application of a Fractional Advection-Dispersion Equation, " Water Resour. Res., 36(6), pp. 1403-1412.
9. Benson, D. A., Wheatcraft, S. W., and Meerschaert, M. M., 2000, "The Fractional-Order Governing Equation of Lévy Motion, " Water Resour. Res., 36(6), pp. 1413-1423.
10. Liu, F., Anh, V, and Turner, I., 2004, "Numerical Solution of the Space Fractional Fokker-Planck Equation, " J. Comput. Appl. Math., 166(1), pp. 209-219.
11. Liu, F., Anh, V., Turner, I., and Zhuang, P., 2003, "Time Fractional Advection-Dispersion Equation, " J. Appl. Math. Computing, 13 (1-2), pp. 233-245.
12. Henry, B. I., Langlands, T. A. M., and Wearne, S. L., 2008, "Fractional Cable Models for Spiny Neuronal Dendrites, " Phys. Rev. Lett., 100, p. 128103.
13. Magin, R., Feng, X., and Baleanu, D., 2009, "Solving the Fractional Order Bloch Equation, " Concepts in Magnetic Resonance Part A, 34 A (1), pp. 16-23.
14. Gorenflo, R., Mainardi, F., Scalas, E., and Raberto, M., 2001, "Fractional Calculus and Continuous-Time Finance. III. The Diffusion Limit, " Mathematical Finance, Birkhäuser-Verlag, Basel, pp. 171-180.
15. Raberto, M., Scalas, E., and Mainardi, F, 2002, "Waiting-Times and Returns in High-Frequency Financial Data: An Empirical Study, " Physica A, 314 (1-4), pp. 749-755.
16. Scalas, E., Gorenflo, R., and Mainardi, F, 2000, "Fractional Calculus and Continuous-Time Finance, " Physica A, 284 (1-4), pp. 376-384.
17. Wyss, W., 2000, "The Fractional Black-Scholes Equation, " Fractional Calculus Appl. Anal., 3, pp. 51-61.
18. Qian, N., and Sejnowski, T., 1989, "An Electro-Diffusion Model for Computing Membrane Potentials and Ionic Concentrations in Branching Dendrites, Spines and Axons, " Biol. Cybern., 62, pp. 1-15.
19. Saxton, M., 2001, "Anomalous Subdiffusion in Fluorescence Photobleaching Recovery: A Monte Carlo Study, " Biophys. J., 81, pp. 2226-2240.
20. Langlands, T. A. M., Henry, B., and Wearne, S., 2005, "Solution of a Fractional Cable Equation: Finite Case, " http://www.maths.unsw.edu.au/ applied/files/2005/amr05-33.pdf
21. Langlands, T. A. M., Henry, B., and Wearne, S., 2005, "Solution of a Fractional Cable Equation: Infinite Case, " http://www.maths.unsw.edu.au/ applied/files/2005/amr05-34.pdf
22. Santamaria, F., Wils, S., De Schutter, E., and Augustine, G. J., 2006, "Anomalous Diffusion in Purkinje Cell Dendrites Caused by Spines, " Neuron, 52, pp. 635-648.
23. Meerschaert, M., and Tadjeran, C, 2004, "Finite Difference Approximations for Fractional Advection-Dispersion Flow Equations, " J. Comput. Appl. Math., 172, pp. 65-77.
24. 2, " J. Comput. Appl. Math., 193, pp. 243-268.
25. Lin, R., and Liu, F, 2007, "Fractional High Order Methods for the Nonlinear Fractional Ordinary Differential Equation, " Nonlinear Anal., 66(4), pp. 856-869.
26. Liu, Q., Liu, F., Turner, I., and Anh, V, 2007, "Approximation of the Levy-Feller Advection-Dispersion Process by Random Walk and Finite Difference Method, " J. Phys.: Condens. Matter, 222, pp. 57-70.
27. Zhuang, P., Liu, F., Anh, V, and Turner, I., 2009, "Numerical Methods for the Variable-Order Fractional Advection-Diffusiion Equation With a Nonlinear Source Term, " SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 47(3), pp. 1760-1761.
28. Lin, R., Liu, F., Anh, V, and Turner, I., 2009, "Stability and Convergence of a New Explicit Finite-Difference Approximation for the Variable-Order Nonlinear Fractional Diffusion Equation, " Applied and Computational Mathematics, 212(2), pp. 435-445.
29. Yu, Q., Liu, F., Anh, V, and Turner, I., 2008, "Solving Linear and Nonlinear Space-Time Fractional Reaction-Diffusion Equations by Adomian Decomposition Method, " Int. J. Numer. Methods Eng., 74, pp. 138-158.
30. Yuste, S. B., and Acedo, L., 2005, "An Explicit Finite Difference Method and a New Von Neumann-Type Stability Analysis for Fractional Diffusion Equations, " SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 42(5), pp. 1862-1874.
31. Yuste, S. B., 2006, "Weighted Average Finite Difference Methods for Fractional Diffusion Equations, " J. Comput. Phys., 216(1), pp. 264-274.
32. Langlands, T. A. M., and Henry, B. I., 2005, "The Accuracy and Stability of an Implicit Solution Method for the Fractional Diffusion Equation, " J. Comput. Phys., 205, pp. 719-736.
33. Chen, C., Liu, F., Turner, I., and Anh, V, 2007, "Fourier Method for the Fractional Diffusion Equation Describing Sub-Diffusion, " J. Comput. Phys., 227, pp. 886-897.
34. Chen, C., Liu, F, and Anh, V, 2009, "A Fourier Method and an Extrapolation Technique for Stokes' First Problem for a Heated Generalized Second Grade Fluid With Fractional Derivative, " J. Comput. Appl. Math., 223, pp. 777-789.
35. Zhuang, P., Liu, F., Anh, V, and Turner, I., 2008, "New Solution and Analytical Techniques of the Implicit Numerical Methods for the Anomalous Sub-Diffusion Equation, " SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal., 46(2), pp. 1079-1095.
36. Liu, F., Yang, C, and Burrage, K., 2009, "Numerical Method and Analytical Technique of the Modified Anomalous Subdiffusion Equation With a Nonlinear Source Term, " J. Comput. Appl. Math., 231(1), pp. 160-176.
37. Oldham, K. B., and Spanier, J., 1974, The Fractional Calculus, Academic, New York.
38. Jacobs, B., Driscoll, L., and Schall, M., 1997, "Life-Span Dendritic and Spine Changes in Areas 10 And 18 of Human Cortex: A Quantitative Golgi Study, " J. Comp. Neurol., 386, pp. 661-680.
39. Duan, H., Wearne, S. L., Rocher, A. B., Macedo, A., Morrison, J. H., and Hof, P. R., 2003, "Age-Related Dendritic and Spine Changes in Corticocortically Projecting Neurons in Macaque Monkeys, " Cereb. Cortex, 13(9), pp. 950-961.
40. Nimchinsky, E. A., Sabatini, B. L., and Svoboda, K., 2002, "Structure and Function of Dendritic Spines, " Annu. Rev. Physiol., 64, pp. 313-353.