Stochastic Boundary Value Problems via Wiener Chaos Expansion

Abstract

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.

Keywords

• stochastic boundary value problems
• Hermite polynomials
• Helmholtz and Navier equation
• weighted Wiener chaos solution