Numerical analysis for a variable-order nonlinear cable equation

Journal of Computational and Applied Mathematics

Volumn 236, Issue 2, 2011, Pages 209-224

  Chen, Chang Ming ^a   Liu, F ^b   Burrage, K ^b,c  

^a XIAMEN UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

Convergence; Fourier analysis; Stability; Variable order nonlinear cable equation; Variable order RiemannLiouville fractional partial derivative

Indexed keywords

CABLE EQUATION; CONVERGENCE; CONVERGENCE AND STABILITY; FIRST-ORDER; FOURTH-ORDER SPATIAL ACCURACY; NUMERICAL EXAMPLE; PARTIAL DERIVATIVES; SECOND ORDERS; VARIABLE-ORDER NONLINEAR CABLE EQUATION;

CABLES; FOURIER ANALYSIS; NONLINEAR ANALYSIS; NONLINEAR EQUATIONS; NUMERICAL ANALYSIS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 80051548248     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cam.2011.06.019     Document Type: Article

References (32)

1. W. Rall Branching dendritic trees and motoneuron membrane resistivity Exp. Neurol. 1 1959 491 527
2. I. Goychuk, and P. Hggi Fractional diffusion modelling of ion channel gating Phys. Rev. E 70 2004 051915
3. B. Henry, T. Langlands, and S. Wearne Fractional cable models for spiny neuronal densites Phys. Rev. Lett. 100 2008 128103
4. B. Hille Ionic Channels of Excitable Membranes 1984 Sinauer Associates Inc. Massachusetts
5. C. Koch Biophysics of Computation, Information Processing in Single Neurons Computational Neuroscience 1999 Oxford University Press New York
6. T.A.M. Langlands, B.I. Henry, and S.L. Wearne Fractional cable equation models for anomalous electrodiffusion in nerve cells: infinite domain solutions J. Math. Biol. 59 2009 761 808
7. T. Langlands, B. Henry, S. Wearne, Solution of a fractional cable equation: finite case, Applied Mathematics Report AMR05/35, University of New South Wales, 2005.
8. T. Langlands, B. Henry, and S. Wearne Anomalous subdiffusion with multispecies linear reaction dynamics Phys. Rev. E 77 2008 021111
9. R. Metzler, and J. Klafter The random walk's guide to anomalous diffusion: A fractional dynamics approach Phys. Rep. 339 2000 1 77
10. N. Qian, and T. Sejnowski An electro-diffusion model for computing membrane potentials and ionic concentrations in branching dendrites spines and axons Biol. Cybernet. 62 1989 1 15 (Pubitemid 20003987)
11. W. Rall Core conductor theory and cable properties of neurons R. Poeter, Handbook of Physiology: The Nervous System vol. 1 1977 American Physiological Society Bethesda, MD 39 97 (Chapter 3)
12. A. Reynolds On the anomalous diffusion characteristics of membrane bound proteins Phys. Lett. A 342 2005 439 442 (Pubitemid 40909800)
13. K. Ritchie Detection of non-Browian diffusion in the cell membrane in single molecure tracking Biophys. J. 88 2005 2266 2277 (Pubitemid 40976233)
14. M. Saxton Anomalous diffusion due to obstacles: A Monte Carlo study Biophys. J. 66 1994 394 401 (Pubitemid 24058459)
15. M. Saxton Anomalous diffusion due to binding: A Monte Carlo study Biophys. J. 70 1996 1250 1262 (Pubitemid 26073038)
16. R. Lin, F. Liu, V. Anh, and I. Turner Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation Appl. Math. Comput. 212 2009 435 445
17. P. Zhuang, F. Liu, V. Anh, and I. Turner Numerical methods for the variable-order fractional advectiondiffusion equation with a nonlinear source term SIAM J. Numer. Anal. 47 2009 1760 1781
18. C.F.M. Coimbra Mechanics with variable-order differential operators Ann. Phys. 12 2003 692 703
19. K.P. Evans, and N. Jacob Feller semigroups obtained by variable order subordination Rev. Mat. Complut. 20 2 2007 293 307
20. N. Jacob, and H. Leopold Pseudo differential operators with variable order of differentiation generating feller semigroup Integral Equations Operator Theory 17 1993 544 553
21. K. Kikuchi, and A. Negoro On Markov processes generated by pseudodifferentail operator of variable order Osaka J. Math. 34 1997 319 335 (Pubitemid 127374957)
22. H.G. Leopold Embedding of function spaces of variable order of differentiation Czechoslovak Math. J. 49 1999 633 644
23. C.F. Lorenzo, T.T. Hartley, Initialization, conceptualization and application in the generalized fractional calculus, NASA/TP-1998-208-208415, 1998.
24. C.F. Lorenzo, and T.T. Hartley Variable-order and distributed order fractional operators Nonlinear Dyn. 29 2002 57 98 (Pubitemid 34945393)
25. L.E.S. Ramirez, and C.F.M. Coimbra Variable order constitutive relation for viscoelas-ticity Ann. Phys. 16 2007 543 552 (Pubitemid 47203902)
26. L.E.S. Ramirez, and C.F.M. Coimbra On the selection and meaning of variable order operators for dynamic modeling Int. J. Differ. 2010 2010 Article ID 846107, 16 pages
27. M.D. Ruiz-Medina, V.V. Anh, and J.M. Angulo Fractional generalized random fields of variable order Stoch. Anal. Appl. 22 2 2004 775 799 (Pubitemid 38613491)
28. S.G. Samko, and B. Ross Integration and differentiation to a variable fractional order Integral Transforms Spec. Funct. 1 1993 277 300
29. C.M. Soon, C.F.M. Coimbra, and M.H. Kobayashi variable viscoelasticity operator Ann. Phys. 14 2005 378 389
30. Chang-Ming Chen, F. Liu, V. Anh, and I. Turner Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation SIAM J. Sci. Comput. 32 4 2010 1740 1760
31. Chang-Ming Chen, F. Liu, V. Anh, and I. Turner Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term Appl. Math. Comput. 217 2011 5729 5742
32. I. Podlubny Fractional Differential Equations 1999 Academic Press New York