On the solution of the Fokker-Planck equation on infinite domains using problem-specific orthonormal basis functions in a Galerkin-type method

Abstract: The dynamical behavior of a mechanical system under random excitation can be described by its probability density function (PDF). The PDF can be determined by random sampling or directly computed as the solution of the corresponding FOKKER-PLANCK equation (FPE). Since the class of FPE with known exact solution is very small, the solution is mostly approximated by numerical methods including finite elements, finite differences or other types of local discretization methods that are best suited for finite domains. Whilst this restriction can be desired for some models, the domain needs to be unbounded in the general case. The authors present a method of calculating a semi-analytical approximate solution for the stationary FPE over an unbounded domain using a GALERKIN-type method. The utilization of a problem-specific orthonormal basis for the ansatz-space ensures that only a small number of improper integrals needs to be computed. In fact, these computations are inexpensive and can be performed prior to the more costly algorithm that constructs the system of linear equations. The computation of the specific basis and its properties are discussed in detail. In order to demonstrate the method and its application on technical systems, the PDF for a nonlinear bistable magneto-piezo-electric energy harvester is calculated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)