A novel high order space-time spectral method for the time fractional fokker-planck equation

SIAM Journal on Scientific Computing

Volumn 37, Issue 2, 2015, Pages A701-A724

  Zheng, Minling ^a   Liu, Fawang ^b   Turner, Ian ^b   Anh, Vo ^b  

^a HUZHOU UNIVERSITY   (China)

^b QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Caputo fractional derivative; Riemann Liouville fractional derivative; Spectral method; Time fractional Fokker Planck equation

Indexed keywords

DIFFERENTIATION (CALCULUS); FOKKER PLANCK EQUATION; INITIAL VALUE PROBLEMS; POLYNOMIALS; SPECTROSCOPY;

BASE FUNCTION; CAPUTO FRACTIONAL DERIVATIVES; FOURIER; FRACTIONAL FOKKER-PLANCK EQUATION; HIGH-ORDER; NUMERICAL SOLUTION; RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES; SPACETIME; SPECTRAL METHODS; TIME FRACTIONAL FOKKE-PLANCK EQUATION;

NUMERICAL METHODS;

EID: 84928954910     PISSN: 10648275     EISSN: 10957197     Source Type: Journal    
DOI: 10.1137/140980545     Document Type: Article

References (49)

1. B. Baeumer, M. Kovács, and M. M. Meerschaert, Numerical solutions for fractional reaction-diffusion equations, Comput. Math. Appl., 55 (2008), pp. 2212-2226.
2. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resourc. Res., 36 (2000), pp. 1403-1412.
3. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, The fractional-order governing equation of Levy motion, Water Resourc. Res., 36 (2000), pp. 1413-1423.
4. P. Biler, T. Funaki, and W. A. Woyczynski, Fractal Burgers equation I, Differ. Equ., 147 (1998), pp. 1-38.
5. L. Blank, Numerical Treatment of Differential Equations of Fractional Order, Numerical Analysis Report 287, Manchester Centre for Computational Mathematics, Manchester, UK, 1996.
6. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods. Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.
7. G.-Q. Chen, Q. Du, and E. Tadmor, Spectral viscosity approximations to multidimensional scalar conservation laws, Math. Comput., 61 (1993), pp. 629-643.
8. C.-M. Chen, F. Liu, and K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput., 198 (2008), pp. 754-769.
9. S. Chen, F. Liu, P. Zhuang, and V. Anh, Finite difference approximations for the fractional Fokker-Planck equation, Appl. Math. Model., 33 (2009), pp. 256-273.
10. W. Deng, Numerical algorithm for the fractional Fokker-Planck equation, J. Comput. Phys., 227 (2007), pp. 1510-1522.
11. W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal., 47 (2008), pp. 204-226.
12. K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electron Trans. Numer. Anal., 5 (1997), pp. 1-6.
13. K. Diethelm, N. Ford, and A. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), pp. 3-22.
14. K. Diethelm and G. Walz, Numerical solution of fractional order differential equations by extroplation, Numer. Algorithms, 16 (1997), pp. 231-253.
15. V. J. Erin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Differential Equations, 22 (2006), pp. 558-576.
16. N. Ford and A. Simpson, The Numerical Solution of Fractional Differential Equations: Speed Versus Accuracy, Numerical Analysis Report 385, Manchester Centre for Computational Mathematics, Manchester, UK, 2001.
17. D. Funaro and D. Gottlieb, A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations, Math. Comput., 51 (1988), pp. 599-613.
18. R. Giona and H. E. Roman, A theory of transport phenomena in disordered systems, Chem. Eng. J., 49 (1992), pp. 1-10.
19. R. Georenflo, Y. Luchko, and F. Mainardi, Wright function as scale-invariant solution of the diffusion wave equation, J. Comput. Appl. Math., 118 (2000), pp. 175-191.
20. T. Y. Hou and R. Li, Computing nearly singular solutions using pseudo-spectral methods, J. Comput. Phys., 226 (2007), pp. 379-397.
21. A. T. 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), pp. 719-736.
22. C.-P. Li, F. Zeng, and F. Liu, Spectral approximations to the fractional integral and derivative, Fract. Calc. Appl. Anal., 15 (2012), pp. 383-406.
23. X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), pp. 2108-2131.
24. Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533-1552.
25. F. Liu, V. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2004), pp. 209-219.
26. F. Liu, V. Anh, I. Turner, and P. Zhuang, Time fractional advection-dispersion equation, J. Appl. Math. Comput., 13 (2003), pp. 233-245.
27. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), pp. 65-77.
28. R. Metzler, E. Barkai, and J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach, Phys. Rev. Lett., 82 (1999), 3563.
29. R. Metzler, E. Barkai, and J. Klafter, Deriving fractional Fokker-Planck equations from a generalised matter equation, Europhys. Lett., 46 (1999), pp. 431-436.
30. R. Metzler and J. Krafter, The random walk's guide to anomolous diffusion: A fractional dynamic approach, Phys. Rep., 239 (2000), pp. 1-72.
31. I. Podlubny, Fractional Differential Equations, Acedemic Press, San Diego, CA, 1999.
32. J. Ren, Z. Sun, and X. Zhao, Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 232 (2013), pp. 456-467.
33. J. P. Roop, Variational Solution of the Fractional Advection-Dispersion Equation, Clemson University, Clemson, SC, 2004.
34. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.
35. W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), pp. 134-144.
36. S. Shen, F. Liu, and V. Anh, Numerical approximations and solution techniques for the spacetime Riesz-Caputo fractional advection-diffusion equation, Numer. Algorithms, 56 (2011), pp. 383-403.
37. J. Shen and T. Tang, Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.
38. J. Shen and L.-L. Wang, Fourierization of the Legendre-Galerkin method and a new spacetime spectral method, Appl. Numer. Math., 57 (2007), pp. 710-720.
39. C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler, A second order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), pp. 205-213.
40. W. Wyss, The fractional diffusion equation, J. Math. Phys., 27 (1986), pp. 2782-2785.
41. Q. Yang, F. Liu, and I. Turner, Numerical solution techniques for time-and space-fractional Fokker-Planck equations, in Proceedings of the 3rd International Federation of Automatic Control Workshop on Fractional Differential and Its Applications, J. Sabatier, R. Magin, B. Vinagre, D. Baleanu, eds., Ankara, Turkey, 2008.
42. S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys., 216 (2006), pp. 264-274.
43. S. B. Yuste and L. Acedo, On an explicit finite difference method for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862-1874.
44. M. Zayernouri and G. E. Karniadakis, Fractional Sturm-Liouville eigen-problems: Theory and numerical approximation, J. Comput. Phys., 252 (2013), pp. 495-517.
45. M. Zayernouri and G. E. Karniadakis, Fractional spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. A40-A62.
46. F. Zeng, C. Li, and F. Liu, I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), pp. A2976-A3000.
47. F. Zeng, F. Liu, C. Li, K. Burrage, I. Turner, and V. Anh, Crank-Nicolson ADI spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation, SIAM J. Numer. Anal., 52 (2014), pp. 2599-2622.
48. H. Zhang, F. Liu, and V. Anh, Galerkin finite element approximation of symmetric spacefractional partial differential equations, Appl. Math. Comput., 217 (2010), pp. 2534-2545.
49. X. Zhao and Z. Sun, A box-type scheme for fractional sub-diffusion equation with Neumann boundary condition, J. Comput. Phys., 230 (2011), pp. 6061-6074.