Journal of Computational and Applied Mathematics
Analysis of some Krylov subspace approximations to the matrix exponential operator
SIAM Journal on Numerical Analysis
Restarted GMRES preconditioned by deflation
Journal of Computational and Applied Mathematics
Deflation Techniques for an Implicitly Restarted Arnoldi Iteration
SIAM Journal on Matrix Analysis and Applications
On Krylov Subspace Approximations to the Matrix Exponential Operator
SIAM Journal on Numerical Analysis
Expokit: a software package for computing matrix exponentials
ACM Transactions on Mathematical Software (TOMS)
Adaptively Preconditioned GMRES Algorithms
SIAM Journal on Scientific Computing
The finite element method using MATLAB (2nd ed.)
The finite element method using MATLAB (2nd ed.)
An unstructured mesh finite volume method for modelling saltwater intrusion into coastal aquifers
The Korean Journal of Computational & Applied Mathematics
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Numerical solution of the space fractional Fokker-Planck equation
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
The accuracy and stability of an implicit solution method for the fractional diffusion equation
Journal of Computational Physics
Analysis of Projection Methods for Rational Function Approximation to the Matrix Exponential
SIAM Journal on Numerical Analysis
Short memory principle and a predictor-corrector approach for fractional differential equations
Journal of Computational and Applied Mathematics
Finite difference/spectral approximations for the time-fractional diffusion equation
Journal of Computational Physics
Pitfalls in fast numerical solvers for fractional differential equations
Journal of Computational and Applied Mathematics
Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions
Journal of Computational Physics
Boundary particle method for Laplace transformed time fractional diffusion equations
Journal of Computational Physics
Journal of Computational Physics
A banded preconditioner for the two-sided, nonlinear space-fractional diffusion equation
Computers & Mathematics with Applications
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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 $_{t}D_{*}^{\gamma}$, $\gamma\in(0,1)$, and the second-order space derivatives with the fractional Laplacian $-(-\Delta)^{\alpha/2}$, $\alpha\in(1,2]$. Using the matrix transfer technique proposed by Ilić et al. [M. Ilić, F. Liu, I. Turner, and V. Anh, Fract. Calc. Appl. Anal., 9 (2006), pp. 333-349], the TSFDE-2D is transformed into a time fractional differential system as $_{t}D_{*}^{\gamma}\mathbf{u} = -K_\alpha\mathbf{A}^{\alpha/2}\mathbf{u}$, where $\mathbf{A}$ is the approximate matrix representation of $(-\Delta)$. Traditional approximation of $\mathbf{A}^{\alpha/2}$ requires diagonalization of $\mathbf{A}$, 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 $f(\mathbf{A})\mathbf{b}$ at each time step, where $\mathbf{b}$ is a suitably defined vector. Depending on the method used to generate the matrix $\mathbf{A}$, the product $f(\mathbf{A})\mathbf{b}$ can be approximated using either the preconditioned Lanczos method when $\mathbf{A}$ is symmetric or the $\mathbf{M}$-Lanzcos method when $\mathbf{A}$ 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.