Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation
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.
Article: New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation[show abstract] [hide abstract]
ABSTRACT: A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous subdiffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the ASub-DE are proposed. The stability and convergence of the INM are investigated by the energy method. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.
Article: Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term[show abstract] [hide abstract]
ABSTRACT: In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.