Analysis of the Laplace Equation on a Circular Sector With a Stochastic Boundary Condition

ABSTRACT We study the Laplace equation with a stochastic boundary condition, defined on a circular domain. We consider two common contexts when dealing with uncertain data, namely, the random partial differential equation and the Itô‐type stochastic partial differential equation. The former is based on regular processes and random parameters, whereas the latter is based upon white Gaussian noise and non‐differentiable trajectories of the solution. As is traditionally done, we focus on the mean‐square solution. The random Fourier series converges in a second‐order sense, which permits rapid approximations for the expectation, the variance, and the covariance.