Stochastic Boundary Value Problems via Wiener Chaos Expansion

In this work, we study stochastic boundary value problems that arise in acoustics and linear elasticity via a Wiener chaos expansion. In particular, for both cases, we provide the appropriate variational formulation for the stochastic-source Helmholtz equation, as well as for the Navier equation wit...

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Bibliographic Details
Published in:Computer sciences & mathematics forum 2023-04, Vol.7 (1), p.34
Main Authors: George Kanakoudis, Konstantinos G. Lallas, Vassilios Sevroglou, Athanasios N. Yannacopoulos
Format: Article
Language:English
Subjects:
Helmholtz and Navier equation
Hermite polynomials
stochastic boundary value problems
weighted Wiener chaos solution
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Summary:In this work, we study stochastic boundary value problems that arise in acoustics and linear elasticity via a Wiener chaos expansion. In particular, for both cases, we provide the appropriate variational formulation for the stochastic-source Helmholtz equation, as well as for the Navier equation with stochastic boundary data. The main idea is to reduce our stochastic problems into an infinite hierarchy of deterministic boundary value problems, for each of which an appropriate variational formulation is considered. Furthermore, we present well-posedness for the above hierarchy of deterministic problems, we give the appropriate linchpin frame with the stochastic problem and we exploit uniqueness and existence arguments for the weighted Wiener chaos solution. Finally, some useful remarks and conclusions are also given.
ISSN:2813-0324
DOI:10.3390/IOCMA2023-14422