A PDE approach to fractional diffusion: a space-fractional wave equation
This work provides a novel numerical framework for solving fractional wave equations, which is important for modeling anomalous diffusion in physics and engineering.
The authors develop a numerical method for a space-fractional wave equation by reformulating it as a quasi-stationary elliptic problem with a dynamic boundary condition on a semi-infinite cylinder, and propose two fully discrete schemes (trapezoidal and leapfrog) with stability and error estimates.
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $Ω$. We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $\mathcal{C} = Ω\times (0,\infty)$. 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 $Ω$ with a suitable $hp$-FEM in the extended variable. For both schemes we derive stability and error estimates.