Finite difference/spectral approximations for the time-fractional diffusion equation

In this paper, we consider the numerical solution of the time fractional diffusion equation. Essentially, the time fractional diffusion equation differs from the standard diffusion equation in the time derivative term. In the former case, the first-order time derivative is replaced by a fractional derivative, making the problem global in time. We propose a spectral method in both temporal and spatial discretizations for this equation. The convergence of the method is proven by providing a priori error estimate. Numerical tests are carried out to confirm the theoretical results. Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.

https://www.jstor.org/stable/info/27862723

A Space-Time Spectral Method for the Time Fractional Diffusion Equation

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

Time fractional diffusion equations are used when attempting to describe transport processes with long memory where the rate of diffusion is inconsistent with the classical Brownian motion model In this paper we develop an implicit unconditionally stable numerical method to solve the one-dimensional linear time fractional diffusion equation, formulated with Caputo's fractional derivative, on a finite slab Several numerical examples of interest are also included

Implicit finite difference approximation for time fractional diffusion equations

Numerical techniques for the variable order time fractional diffusion equation

In this paper, we consider a space-time Riesz---Caputo fractional advection-diffusion equation. The equation is obtained from the standard advection-diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order ????(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of order β 1???(0,1) and β 2???(1,2], respectively. We present an explicit difference approximation and an implicit difference approximation for the equation with initial and boundary conditions in a finite domain. Using mathematical induction, we prove that the implicit difference approximation is unconditionally stable and convergent, but the explicit difference approximation is conditionally stable and convergent. We also present two solution techniques: a Richardson extrapolation method is used to obtain higher order accuracy and the short-memory principle is used to investigate the effect of the amount of computations. A numerical example is given; the numerical results are in good agreement with theoretical analysis.

Numerical approximations and solution techniques for the space-time Riesz---Caputo fractional advection-diffusion equation

In this paper, we consider a Riesz fractional advection-dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order a e (0, 1) and fi e (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection-dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection-dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation

The space fractional diffusion equation (?) is obtained from the classical diffusion equation by replacing the second space derivative by a fractional derivative of order ?, 1 ? ? ? 2 . Numerical methods associated with integer-order differential equation, have been extensively treated. On the other hand, studies of the numerical methods and error estimates of fractional order differential equations are quite limited to date. Here, we propose an explicit finite difference approximation (?) for ?. An error analysis of the explicit numerical method for ? with insulated ends is discussed. We derive the scaling restriction of the stability and convergence of the explicit numerical method. Finally, some numerical results show the diffusion behaviour according to the order of space-fractional derivative and demonstrate that our ? is computationally simple for ?.

/pdf/error-analysis-of-an-explicit-finite-difference-59rje1hgcf.pdf

Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends

The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order in (0,1). In this work, an explicit finite-difference scheme for TFDE is presented. Discrete models of a non-Markovian random walk are generated for simulating random processes whose spatial probability density evolves in time according to this fractional diffusion equation. We derive the scaling restriction of the stability and convergence of the discrete non-Markovian random walk approximation for TFDE in a bounded domain. Finally, some numerical examples are presented to show the application of the present technique.

/pdf/analysis-of-a-discrete-non-markovian-random-walk-57xmq1kml1.pdf

Shujun Shen

Papers

Numerical techniques for the variable order time fractional diffusion equation

Numerical approximations and solution techniques for the space-time Riesz---Caputo fractional advection-diffusion equation

The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation

Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends

Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation