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# Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, QLD 4001, Australia; School of Mathematical Sciences, Xiamen University, Xiamen 361006, China; Department of Mathematics, University of Queensland, QLD 4072, Australia
Applied Mathematics and Computation DOI:10.1016/j.amc.2006.08.162

ABSTRACT In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0, 1], and the first-order and second-order space derivatives by the Riemman–Liouville fractional derivatives of order β ∈ (0, 1] and of order γ ∈ (1, 2], respectively. For the space fractional derivatives and , we adopted the Grünwald formula and the shift Grünwald formula, respectively. We propose an implicit difference method (IDM) and an explicit difference method (EDM) to solve this equation. Stability and convergence of these methods are discussed. Using mathematical induction, we prove that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and convergent. Numerical results are in good agreement with theoretical analysis.

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### Keywords

Caputo fractional derivative

convergence

EDM

explicit difference method

finite domain

good agreement

Grünwald formula

implicit difference method

mathematical induction

order α ∈

order β ∈

order γ ∈

Riemman–Liouville fractional derivatives

second-order space derivatives

shift Grünwald formula

space fractional derivatives