期刊
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
卷 15, 期 4, 页码 1063-1098出版社
GLOBAL SCIENCE PRESS
DOI: 10.4208/nmtma.OA-2022-0013s
关键词
Galerkin finite element method; semilinear stochastic time-tempered fractional wave equation; fractional Laplacian; multiplicative Gaussian noise; additive fractional Gaussian noise
资金
- National Natural Science Foundation of China [41875084, 11801452, 12071195, 12225107]
- AI and Big Data Funds [2019620005000775]
- Innovative Groups of Basic Research in Gansu Province [22JR5RA391]
- NSF of Gansu [21JR7RA537]
This paper studies the model of wave propagation in inhomogeneous media with frequency dependent power-law attenuation, and proposes a Galerkin finite element approximation for the semilinear stochastic fractional wave equation. By discretizing the multiplicative Gaussian noise and fractional Gaussian noise, a regularized stochastic fractional wave equation is obtained, and the modeling error and approximation error are estimated. Numerical experiments are performed to validate the theoretical analysis.
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.
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