Numerical techniques for the variable order time fractional diffusion equation

Applied Mathematics and Computation

Volumn 218, Issue 22, 2012, Pages 10861-10870

  Shen, S ^a   Liu, F ^b   Chen, J ^c   Turner, I ^b   Anh, V ^b  

^a HUAQIAO UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

^c JIMEI UNIVERSITY   (China)

Author keywords

Convergence; Fourier analysis; Numerical method; Solvability; Stability; Variable order time fractional diffusion equation

Indexed keywords

COMPUTATIONALLY EFFICIENT; CONVERGENCE; FRACTIONAL OPERATORS; FUNDAMENTAL CHARACTERISTICS; NUMERICAL EXAMPLE; NUMERICAL SCHEME; NUMERICAL TECHNIQUES; PHYSICAL MODELING; PHYSICAL VARIABLES; SOLVABILITY; TIME FRACTIONAL DIFFUSION EQUATION; VARIABLE ORDER;

CONVERGENCE OF NUMERICAL METHODS; FOURIER ANALYSIS; MODELS; NUMERICAL METHODS;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 84862850103     PISSN: 00963003     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.amc.2012.04.047     Document Type: Article

References (29)

1. V.V. Anh, and N.N. Leonenko Spectral analysis of fractional kinetic equations with random data J. Statis. Phys. 104 5-6 2001 1349 1387
2. T. Blaszczyk, M. Ciesielski, M. Klimek, and J. Leszczynski Numerical solution of fractional oscillator equation Appl. Math. Comput. 218 2011 2480 2488
3. C. 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
4. C.F.M. Coimbra Mechanics with variable-order differential operators Ann. Phys. 12 11-12 2003 692 703
5. G. Diaz, and C.F.M. Coimbra Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation Nonlinear Dynam. 56 2009 145 157
6. Leda Galue, S.L. Kalla, and B.N. Al-Saqabi Fractional extensions of the temperature field problems in oil strata Appl. Math. Comput. 186 2007 35 44
7. J.M. Holte, Discrete Gronwall lemma and applications, MAA-NCS Meeting at the University of North Dakota, 24 October 2009.
8. D. Ingman, J. Suzdalnitsky, and M. Zeifman Constitutive dynamic-order model for nonlinear contact phenomena J. Appl. Mech. 67 2000 383 390
9. D. Ingman, and J. Suzdalnitsky Control of damping oscilations by fractional differential operator with time-dependent order Comput. Methods Appl. Mech. Eng. 193 2004 5585 5595
10. T.A.M. Langlands, and B.I. Henry The accuracy and stability of an implicit solution method for the fractional diffusion equation J. Comput. Phys. 205 2005 719 736
11. R. Lin, F. Liu, V. Anh, and I. Turner Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractonal diffusion equation Appl. Math. Comput. 212 2009 435 445
12. Y. Lin, and C. Xu Finite difference/spectral approximations for the time-fractional diffusion equation J. Comput. Phys. 225 2007 1533 1552
13. F. Liu, V. Anh, and I. Turner Numerical solution of space fractional Fokker-Planck equation J. Comput. Appl. Math. 166 2004 209 219
14. F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage Stability and convergence of difference methods for the space-time fractional advection-diffusion equation Appl. Math. Comput. 191 1 2007 12 20
15. F. Liu, C. Yang, and K. Burrage Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term J. Comput. Appl. Math. 231 2009 160 176
16. F. Liu, and K. Burrage Novel techniques in parameter estimation for fractional dynamical models arising from biological systems Comput. Math. Appl. 62 2011 822 833
17. F. Liu, Q. Yang, and I. Turner Two new implicit numerical methods for the fractional cable equation J. Comput. Nonlinear Dynam. 6 1 2011 01108 10.1115/1.4002267
18. F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Comput. Math. Appl., in press. .
19. C.F. Lorenzo, and T.T. Hartley Initialization, conceptualization, and application in the generalized fractional calculus, NASA Technical Publication 98-208415 1998 NASA Lewis Reseach Center
20. C.F. Lorenzo, and T.T. Hartley Variable order and distributed order fractional operators Nonlinear Dynam. 29 2002 57 98
21. Zaid Odibat Approximations of fractional integrals and Caputo fractional derivatives Appl. Math. Comput. 178 2006 527 533
22. H.T.C. Pedro, M.H. Kobayashi, J.M.C. Pereira, and C.F.M. Coimbra Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere J. Vib. Control 14 2008 1659 1672
23. E.S. Ramirez, and F.M. Coimbra On the selection and meaning of variable order operators for dynamic modeling Int. J. Differ. Equat. Vol. 2010 10.1155/2010/846107
24. S.G. Samko, and B. Ross Intergation and differentiation to a variable fractional order Integral Trans. Special Func. 1 4 1993 277 300
25. S.G. Samko Fractional integration and differentiation of variable order Anal. Math. 21 1995 213 236
26. C.M. Soon, F.M. Coimbra, and M.H. Kobayashi The variable viscoelasticity oscillator Ann. Phys. 14 6 2005 378 389
27. H. Sun, W. Chen, and Y. Chen Variable-order fractional differential operators in anomalous diffusion modeling Physica A 388 2009 4586 4592
28. P. Zhuang, F. Liu, V. Anh, and I. Turner New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation SIAM J. Numer. Anal. 46 2 2008 1079 1095
29. P. Zhuang, F. Liu, V. Anh, and I. Turner Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term SIAM J. Numer. Anal. 47 2009 1760 1781