The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems (Communications in Computational Physics, 2011)
The probabilistic solutions of the nonlinear stochastic dynamic (NSD) systemswith polynomial type of nonlinearity are investigated with the subspace-EPCmethod. The space of the state variables of large-scale nonlinear stochastic dynamicsystem excited by white noises is separated into two subspaces. Both sides of theFokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is thenintegrated over one of the subspaces. The FPK equation for the joint probability densityfunction of the state variables in another subspace is formulated. Therefore, theFPK equation in low dimensions is obtained from the original FPK equation in highdimensions and it makes the problem of obtaining the probabilistic solutions of large-scaleNSD systems solvable with the exponential polynomial closure method. Examplesabout the NSD systems with polynomial type of nonlinearity are given to showthe effectiveness of the subspace-EPC method in these cases.