Commun. Comput. Phys., 10 (2011), pp. 1241-1256. |
The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems Guo-Kang Er ^{1*}, Vai Pan Iu ^{1} 1 Faculty of Science and Technology, University of Macau, Macau SAR, P. R. China.Received 14 July 2010; Accepted (in revised version) 21 January 2011 Available online 2 August 2011 doi:10.4208/cicp.140710.210111a Abstract The probabilistic solutions of the nonlinear stochastic dynamic (NSD) systems with polynomial type of nonlinearity are investigated with the subspace-EPC method. The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system is then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in another subspace is formulated. Therefore, the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of large-scale NSD systems solvable with the exponential polynomial closure method. Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases. Notice: Undefined variable: ams in /var/www/html/issue/abstract/readabs.php on line 163 PACS: 05.10.Gg, 05.10.-a, 02.30.Jr Key words: Nonlinear stochastic dynamic systems, large-scale systems, probability density function, Fokker-Planck-Kolmogorov equation, subspace. *Corresponding author. Email: gker@umac.mo (G.-K. Er), vaipaniu@umac.mo (V. P. Iu) |