GALERKIN FINITE ELEMENT APPROXIMATIONS FOR STOCHASTIC SPACE-TIME FRACTIONAL WAVE EQUATIONS

摘要

The traditional wave equation models wave propagation in an ideal medium. To characterize wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, the space-time fractional wave equation is needed; further incorporating the additive white Gaussian noise (external noise) coming from many natural sources leads to the stochastic space-time fractional wave equation. This paper discusses the Galerkin finite element approximations for the stochastic space-time fractional wave equation forced by an additive space-time white noise. We first discretize the space-time additive noise, which introduces a modeling error and results in a regularized stochastic space-time fractional wave equation; then the regularity of the regularized equation is analyzed. For the discretization in space, the finite element approximation is used and the definition of the discrete fractional Laplacian is introduced. We derive the mean-squared L-2-norm a priori estimates for the modeling error and for the approximation error to the solution of the regularized problem; numerical experiments are then performed to confirm the estimates. For the time-stepping, we calculate the analytical solution of the space semidiscretized equation, which is expressed by Mittag-Leffler functions.