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Numerical Analysis and Simulation of Random Partial Differential Equations with Boundary Excitations
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Important physical phenomena can be described in terms of partial differential equations that are perturbed by random forces acting on the boundaries of the domain under consideration, like the ground-motion excitation of multi-story buildings or the top-layer thermal excitation of the earth’s atmosphere. In particular, many of these random forces are given as filtered white noise processes (colored noise) or power law noises. This leads to the mathematical study of random partial differential equations (RPDEs) with path-wise continuous or path-wise differentiable solution stochastic processes. From the numerical point of view RPDEs are simulated by polynomial chaos methods or stochastic Galerkin schemes. Here, we focus on the theoretical and numerical properties of a different approach tailored for boundary excited problems: a method of lines that reduces RPDEs to a finite-dimensional system of random ordinary differential equations (RODEs). Novel numerical schemes allow the effective and efficient computation of solutions of RODEs such that this reduction provides a very practical alternative to traditional numerical schemes for boundary excited RPDEs. We illustrate our RPDE-RODE reduction method with the aid of ground-motion induced oscillations of solid structures where we discuss existence of weak solution as well as the convergence properties of the numerical scheme.