K. Mustapha

K. Mustapha, K. Furati, O. M. Knio et al.

Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathematical analysis of these models and the establishment of suitable numerical schemes. This paper proposes and analyzes the first finite difference method for solving {\em variable-coefficient} fractional differential equations, with two-sided fractional derivatives, in one-dimensional space. The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided fractional derivative when the right-sided fractional derivative is approximated by two consecutive applications of the first-order backward Euler method. Our finite difference scheme reduces to the standard second-order central difference scheme in the absence of fractional derivatives. The existence and uniqueness of the solution for the proposed scheme are proved, and truncation errors of order $h$ are demonstrated, where $h$ denotes the maximum space step size. The numerical tests illustrate the global $O(h)$ accuracy of our scheme, except for nonsmooth cases which, as expected, have deteriorated convergence rates.

K. Mustapha, B. Abdallah, K. M. Furati et al.

We propose a piecewise-linear, time-stepping discontinuous Galerkin method to solve numerically a time fractional diffusion equation involving Caputo derivative of order $μ\in (0,1)$ with variable coefficients. For the spatial discretization, we apply the standard piecewise linear continuous Galerkin method. Well-posedness of the fully discrete scheme and error analysis will be shown. For a time interval~$(0,T)$ and a spatial domain~$Ω$, our analysis suggest that the error in $L^2\bigr((0,T),L^2(Ω)\bigr)$-norm is of order $O(k^{2-\fracμ{2}}+h^2)$ (that is, short by order $\fracμ{2}$ from being optimal in time) where $k$ denotes the maximum time step, and $h$ is the maximum diameter of the elements of the (quasi-uniform) spatial mesh. However, our numerical experiments indicate optimal $O(k^{2}+h^2)$ error bound in the stronger $L^\infty\bigr((0,T),L^2(Ω)\bigr)$-norm. Variable time steps are used to compensate the singularity of the continuous solution near $t=0$.