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)
Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions. (English) Zbl 1229.35315
Summary: In this paper, a time-space fractional diffusion equation in two dimensions (TSFDE-2D) with homogeneous Dirichlet boundary conditions is considered. The TSFDE-2D is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo fractional derivative t D * γ , γ(0,1), and the second-order space derivatives with the fractional Laplacian -(-Δ) α/2 , α(1,2]. Using the matrix transfer technique proposed by M. Ilić et al. [Fract. Calc. Appl. Anal. 9, No. 4, 333–349 (2006; Zbl 1132.35507)], the TSFDE-2D is transformed into a time fractional differential system as t D * γ 𝐮=-K α 𝐀 α/2 𝐮, where 𝐀 is the approximate matrix representation of (-Δ). Traditional approximation of 𝐀 α/2 requires diagonalization of 𝐀, which is very time-consuming for large sparse matrices. The novelty of our proposed numerical schemes is that, using either the finite difference method or the Laplace transform to handle the Caputo time fractional derivative, the solution of the TSFDE-2D is written in terms of a matrix function vector product f(𝐀)𝐛 at each time step, where 𝐛 is a suitably defined vector. Depending on the method used to generate the matrix 𝐀, the product f(𝐀)𝐛 can be approximated using either the preconditioned Lanczos method when 𝐀 is symmetric or the 𝐌-Lanzcos method when 𝐀 is nonsymmetric, which are powerful techniques for solving large linear systems. We give error bounds for the new methods and illustrate their roles in solving the TSFDE-2D. We also derive the analytical solution of the TSFDE-2D in terms of the Mittag-Leffler function. Finally, numerical results are presented to verify the proposed numerical solution strategies.
MSC:
35R11Fractional partial differential equations
26A33Fractional derivatives and integrals (real functions)
65F60Matrix exponential and similar matrix functions (numerical linear algebra)
65M06Finite difference methods (IVP of PDE)