Correlation Structure of the Solution to the Reaction-Diffusion Equation in Respond to Random Fluctuations of the Boundary Conditions

Abstract

In this paper, we deal with the reaction-diffusion equation subject to Dirichlet and Neumann boundary conditions where the input function on the boundary is randomly fluctuated. First we study the fundamental case when this function is a white noise. Explicit form of the correlation function is derived for the reaction-diffusion equation in a half-plane. In this case we obtain the Karhunen–Loève expansion (KL) of the solution which is a partially homogeneous random field, i.e., it is homogeneous along the horizontal direction, and is inhomogeneous in the vertical direction. Then, based on this representation, we extend this result to the general case when the function prescribed on the boundary is an arbitrary homogeneous random field.