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Volume 10, Issue 5
The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems

Guo-Kang Er & Vai Pan Iu

DOI: 10.4208/cicp.140710.210111a

Commun. Comput. Phys., 10 (2011), pp. 1241-1256.

Published online: 2011-10

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  • 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.

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@Article{CiCP-10-1241, author = {Guo-Kang Er and Vai Pan Iu}, title = {The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {5}, pages = {1241--1256}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.140710.210111a}, url = {http://global-sci.org/intro/article_detail/cicp/7482.html} }
TY - JOUR T1 - The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems AU - Guo-Kang Er & Vai Pan Iu JO - Communications in Computational Physics VL - 5 SP - 1241 EP - 1256 PY - 2011 DA - 2011/10 SN - 10 DO - http://doi.org/10.4208/cicp.140710.210111a UR - https://global-sci.org/intro/article_detail/cicp/7482.html KW - AB -

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.

Guo-Kang Er and Vai Pan Iu. (2011). The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems. Communications in Computational Physics. 10 (5). 1241-1256. doi:10.4208/cicp.140710.210111a
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