TY - JOUR
T1 - Galerkin Finite Element Approximation for Semilinear Stochastic Time-Tempered Fractional Wave Equations with Multiplicative Gaussian Noise and Additive Fractional Gaussian Noise
AU - Li , Yajing
AU - Wang , Yejuan
AU - Deng , Weihua
AU - Nie , Daxin
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 4
SP - 1063
EP - 1098
PY - 2022
DA - 2022/10
SN - 15
DO - http://doi.org/10.4208/nmtma.OA-2022-0013s
UR - https://global-sci.org/intro/article_detail/nmtma/21090.html
KW - Galerkin finite element method, semilinear stochastic time-tempered fractional wave equation, fractional Laplacian, multiplicative Gaussian noise, additive fractional Gaussian noise.
AB - <p style="text-align: justify;">To model wave propagation in inhomogeneous media with frequency dependent 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 Gaussian noise
and additive 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. First,
the space-time multiplicative Gaussian noise and additive 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 a mean-square priori estimates
for the modeling error and the approximation error to the solution of the regularized
problem are established. Finally, numerical experiments are performed to confirm
the theoretical analysis.</p>