Stochastic Boundary Value Problems via Wiener-Chaos Expansion

In this work, we study stochastic problems in acoustics as well as in linear elasticity in two dimensions. We first provide the appropriate variational formulation for the stochastic source Helmholtz equation. By using Wiener-chaos expansion for the stochastic source, we transform the stochastic problem in a hierarchy of deterministic boundary value problems. The latter will be applied in order to establish existence and uniqueness for the finite element approximate solution of the stochastic source Helmholtz equation. Further, following the same concept of decomposing a stochastic problem into an hierarchy of deterministic ones we present an elastic scattering Dirichlet problem where the medium density is a random variable. In particular we consider a modified Navier equation introducing the density with its Wiener-chaos expansion in a combination with the appropriate Wick product. We reconstruct our solution via Wiener-chaos expansion techniques and finally we give some useful results and conclusions.