Galerkin finite element approximation for semilinear stochastic time-tempered fractional wave equations with multiplicative white noise and fractional Gaussian noise

Abstract

To model wave propagation in inhomogeneous media with frequencydependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative white noise and fractional Gaussian noise, because of the potential fluctuations of the external sources. The purpose of this work is to discuss the Galerkin finite element approximation for the semilinear stochastic fractional wave equation. We first provide a complete solution theory, e.g., existence, uniqueness, and regularity. Then the space-time multiplicative white noise and fractional Gaussian noise are discretized, which results in a regularized stochastic fractional wave equation while introducing a modeling error in the mean-square sense. We further present a complete regularity theory for the regularized equation. A standard finite element approximation is used for the spatial operator, and the mean-square priori estimates for the modeling error and for the approximation error to the solution of the regularized problem are established.

Numerical analysis for stochastic time-space fractional diffusion equation driven by fractional Gaussion noise

“…Extensive numerical schemes for the deterministic fractional diffusion equation (1.3) have been proposed in [2, 5,28]. Also, there have been many works for numerically solving stochastic partial differential equations (PDEs) involving Laplace and spectral fractional Laplacian; one can refer to [7,16,17,20,21,23,29,33]. But for stochastic PDEs involving integral fractional Laplacian, the related researches are still few.…”

Strong convergence order for the scheme of fractional diffusion equation driven by fractional Gaussion noise

“…As for the numerical approximations of the time fractional partial differential equations driven by fractional Gaussian noise, there are relatively few studies. Reference [15] presents the Galerkin finite element semi-discrete scheme for semilinear stochastic time-tempered fractional wave equations driven by multiplicative Gaussian noise and fractional Gaussian noise and provides the error analyses when the noise is regular enough.…”

“…In this paper, we use backward Euler convolution quadrature and finite element method to discretize the time fractional derivative and the space operator, respectively. Different from the Gaussian noise, fractional Gaussian noise is more regular in time, that is, the trajectory of fractional Brownian motion belongs to C H ([0, T ]), so the corresponding estimate depends on the Hurst index H. But the approach adopted in [15,17] can't reflect the influence of Hurst index on regularity of the solutions and convergence rates. Here, we take the appropriate weighted function and combine the operator theoretical approach to get the regularity estimate for the solution, i.e., (…”