Finite Difference Approximation of Fractional Wave Equation with Concentrated Capacity

Delić, Aleksandra; Jovanović, Boško S.

doi:10.1515/cmam-2016-0026

Cited by 4 publications

(1 citation statement)

References 14 publications

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“…There are already some discussions on the numerical methods or correct ways of specifying the boundary conditions for tempered fractional differential equations; see, e.g., [2,17,21,44,45] and the references therein or [15,16]. As for the fractional wave equations, there are also some progresses not only on their numerical methods [13,14,19,22,39,43] but also on their fundamental solutions and properties [6,20,25,37].…”

Section: Introductionmentioning

confidence: 99%

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

Wang¹,

Deng²

2019

Preprint

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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.

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