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To model wave propagation in inhomogeneous media with frequency-dependentpower-law attenuation, it is needed to use the fractional powers of symmetriccoercive elliptic operators in space and the Caputo tempered fractionalderivative in time. The model studied in this paper is semilinear stochasticspace-time fractional wave equations driven by infinite dimensionalmultiplicative white noise and fractional Gaussian noise, because of thepotential fluctuations of the external sources. The purpose of this work is todiscuss the Galerkin finite element approximation for the semilinear stochasticfractional wave equation. We first provide a complete solution theory, e.g.,existence, uniqueness, and regularity. Then the space-time multiplicative whitenoise and fractional Gaussian noise are discretized, which results in aregularized stochastic fractional wave equation while introducing a modelingerror in the mean-square sense. We further present a complete regularity theoryfor the regularized equation. A standard finite element approximation is usedfor the spatial operator, and the mean-square priori estimates for the modelingerror and for the approximation error to the solution of the regularizedproblem are established.

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