An integral transform approach for solving partial differential equations with stochastic excitation

Abstract

In this paper an integral transform approach for solving a class of initial-boundary-value problems involving linear stochastic partial differential equations (SPDEs) encountered in engineering mechanics applications is presented. The stochastic solution is represented in the complex z-plane, and is inspired by the corresponding approach to deterministic PDEs already established in the literatures. In this regard, it is noted that the determination of relevant correlation function and spectral density of the solution is expedited because of the induced separability of the displacementx, and the timetin their representations. For three kinds of SPDEs on the half-line {0<x<∞,t>0} involving a spatially dependent white noise excitation, relevant response statistics are determined. The influence of the boundary condition on the correlation function of the response is also discussed. Further, the corresponding spectral density is also expressed as an integral in the complex z-plane using the Wiener–Khintchine relation. Numerical examples pertaining to the stochastic heat equation suggest that the new transform approach is a viable tool of analysis.