The Random Steady State Diffusion Problem. I: Random Generalized Solutions to Laplace’s Equation
Georges A. Bécus, Francis A. Cozzarelli · SIAM Journal on Applied Mathematics · 1976
The following three problems for Laplace’s equation are considered: (i) generalized Dirichlet problem for a random boundary function, (ii) generalized Poisson problem for a random right-hand side, and (iii) generalized Dirichlet–Poisson problem for random boundary function and right-hand side. For each of these problems a solution which is an extension of both the deterministic solution and Babuška’s results is defined. Existence, uniqueness, and conditions under which such solutions are equivalent to both sample and mean square solutions are established. In each case the properties of the solution are analyzed.