zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Finite difference/spectral approximations for the time-fractional diffusion equation. (English) Zbl 1126.65121
Consider the problem of finding a numerical solution for the time-fractional diffusion equation with the fractional derivative with respect to time being given in the sense of Caputo. For the solution of this problem, the authors propose a method based on a finite difference scheme with respect to time combined with a Legendre spectral method for the space variable(s). A proof of stability and convergence of the algorithm is provided. Error estimates and numerical results are given as well.

MSC:
65R20Integral equations (numerical methods)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
65M70Spectral, collocation and related methods (IVP of PDE)
45K05Integro-partial differential equations
26A33Fractional derivatives and integrals (real functions)
35K15Second order parabolic equations, initial value problems
WorldCat.org
Full Text: DOI
References:
[1] Agrawal, O. P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. J. nonlinear dynam. 29, 145-155 (2002) · Zbl 1009.65085
[2] Berberan-Santos, M. N.: Properties of the Mittag -- Leffler relaxation function. J. math. Chem. 38, No. 4, 629-635 (2005) · Zbl 1101.33015
[3] Bernardi, C.; Maday, Y.: Approximations spectrales de problems aux limites elliptiques. (1992) · Zbl 0773.47032
[4] Diethelm, K.; Ford, N. J.: Numerical solution of the bagley -- torvik equation. Bit 42, 490-507 (2002) · Zbl 1035.65067
[5] Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. Proceedings of the second conference on scientific computing in chemical engineering, 217-224 (1999)
[6] Deng, Z.; Singh, V. P.; Bengtsson, L.: Numerical solution of fractional advection -- dispersion equation. J. hydraulic eng. 130, 422-431 (2004)
[7] Fix, G. J.; Roop, J. P.: Least squares finite element solution of a fractional order two-point boundary value problem. Comput. math. Appl. 48, 1017-1033 (2004) · Zbl 1069.65094
[8] Gorenflo, R.; Luchko, Y.; Mainardi, F.: Wright function as scale-invariant solutions of the diffusion-wave equation. J. comp. Appl. math. 118, 175-191 (2000) · Zbl 0973.35012
[9] Gorenflo, R.; Mainardi, F.; Moretti, D.; Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear dynam. 29, 129-143 (2002) · Zbl 1009.82016
[10] Huang, F.; Liu, F.: The time fractional diffusion and advection -- dispersion equation. Anziam j. 46, 1-14 (2005)
[11] Liu, F.; Anh, V.; Turner, I.; Zhuang, P.: Time fractional advection dispersion equation. J. appl. Math. comput. 13, 233-245 (2003) · Zbl 1068.26006
[12] Liu, F.; Shen, S.; Anh, V.; Turner, I.: Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation. Anziam j. 46, No. E, 488-504 (2005)
[13] Lynch, V. E.; Carreras, B. A.; Del-Castillo-Negrete, D.; Ferreira-Mejias, K. M.; Hicks, H. R.: Numerical methods for the solution of partial differential equations of fractional order. J. comput. Phys. 192, 406-421 (2003) · Zbl 1047.76075
[14] Mainardi, F.: Some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004
[15] Meerschaert, M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. comp. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346
[16] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[17] Quarteroni, A.; Valli, A.: Numerical approximation of partial differential equations. (1994) · Zbl 0803.65088
[18] Schneider, W. R.; Wyss, W.: Fractional diffusion and wave equations. J. math. Phys. 30, 34-144 (1989) · Zbl 0692.45004
[19] Tang, T.: A finite difference scheme for partial integro-differential equations with a weakly singular kernel. Appl. numer. Math. 11, No. 4, 309-319 (1993) · Zbl 0768.65093
[20] Tadjeran, C.; Meerschaert, M.; Scheffler, H. P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. comput. Phys. 213, 205-213 (2006) · Zbl 1089.65089
[21] Wyss, W.: The fractional diffusion equation. J. math. Phys. 27, 2782-2785 (1986) · Zbl 0632.35031
[22] Xu, C. J.; Lin, Y. M.: Analysis of iterative methods for the viscous/inviscid coupled problem via a spectral element approximation. Inter. J. Numer. met. Fluids 32, 619-646 (2000) · Zbl 0981.76066