Now showing items 1-3 of 3

• #### An algorithm for the numerical solution of two-sided space-fractional partial differential equations.

We introduce an algorithm for solving two-sided space-fractional partial differential equations. The space-fractional derivatives we consider here are left-handed and right-handed Riemann–Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. We approximate the Hadamard finite-part integrals by using piecewise quadratic interpolation polynomials and obtain a numerical approximation of the space-fractional derivative with convergence order
• #### Some time stepping methods for fractional diffusion problems with nonsmooth data

We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an $O(k)$ convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator $A$ is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where $k$ denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or $L1$ scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate $O(k)$ with nonsmooth initial data for the homogeneous problem when $A$ is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is $O(k^{1+ \alpha}), 0<\alpha <1$ with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is $O(k^{1+ \alpha}), 0<\alpha <1$ with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results.
• #### Stability of a numerical method for a fractional telegraph equation

In this paper, we introduce a numerical method for solving the time-space fractional telegraph equations. The numerical method is based on a quadrature formula approach and a stability condition for the numerical method is obtained. Two numerical examples are given and the stability regions are plotted.