16,870 research outputs found
Second order finite difference approximations for the two-dimensional time-space Caputo-Riesz fractional diffusion equation
In this paper, we discuss the time-space Caputo-Riesz fractional diffusion
equation with variable coefficients on a finite domain. The finite difference
schemes for this equation are provided. We theoretically prove and numerically
verify that the implicit finite difference scheme is unconditionally stable
(the explicit scheme is conditionally stable with the stability condition
) and 2nd order convergent in space direction, and
-th order convergent in time direction, where .Comment: 27 page
A PDE approach to fractional diffusion: a space-fractional wave equation
We study solution techniques for an evolution equation involving second order
derivative in time and the spectral fractional powers, of order ,
of symmetric, coercive, linear, elliptic, second-order operators in bounded
domains . We realize fractional diffusion as the Dirichlet-to-Neumann
map for a nonuniformly elliptic problem posed on the semi-infinite cylinder
. We thus rewrite our evolution problem
as a quasi-stationary elliptic problem with a dynamic boundary condition and
derive space, time, and space-time regularity estimates for its solution. The
latter problem exhibits an exponential decay in the extended dimension and thus
suggests a truncation that is suitable for numerical approximation. We propose
and analyze two fully discrete schemes. The discretization in time is based on
finite difference discretization techniques: trapezoidal and leapfrog schemes.
The discretization in space relies on the tensorization of a first-degree FEM
in with a suitable -FEM in the extended variable. For both schemes
we derive stability and error estimates
An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations
In this paper, we consider the initial boundary value problem of the two
dimensional multi-term time fractional mixed diffusion and diffusion-wave
equations. An alternating direction implicit (ADI) spectral method is developed
based on Legendre spectral approximation in space and finite difference
discretization in time. Numerical stability and convergence of the schemes are
proved, the optimal error is , where are the
polynomial degree, time step size and the regularity of the exact solution,
respectively. We also consider the non-smooth solution case by adding some
correction terms. Numerical experiments are presented to confirm our
theoretical analysis. These techniques can be used to model diffusion and
transport of viscoelastic non-Newtonian fluids
- …