New partial differential equations governing the joint, response-excitation, probability distributions of nonlinear systems, under general stochastic excitation

Sapsis, TP; Athanassoulis, GA

dc.contributor.author	Sapsis, TP	en
dc.contributor.author	Athanassoulis, GA	en
dc.date.accessioned	2014-03-01T01:28:51Z
dc.date.available	2014-03-01T01:28:51Z
dc.date.issued	2008	en
dc.identifier.issn	0266-8920	en
dc.identifier.uri	https://dspace.lib.ntua.gr/xmlui/handle/123456789/18998
dc.subject	Characteristic functional	en
dc.subject	Correlated stochastic excitation	en
dc.subject	Functional Differential equations	en
dc.subject	Generalized Fokker-Planck-Kolmogorov equation	en
dc.subject	Kernel density functions	en
dc.subject	Non-Markovian responses	en
dc.subject	Stochastic differential equations	en
dc.subject	Stochastic dynamics	en
dc.subject.classification	Engineering, Mechanical	en
dc.subject.classification	Mechanics	en
dc.subject.classification	Statistics & Probability	en
dc.subject.other	Control nonlinearities	en
dc.subject.other	Nonlinear systems	en
dc.subject.other	Numerical methods	en
dc.subject.other	Polynomials	en
dc.subject.other	Probability distributions	en
dc.subject.other	Random processes	en
dc.subject.other	Characteristic functional	en
dc.subject.other	Correlated stochastic excitation	en
dc.subject.other	Functional Differential equations	en
dc.subject.other	Generalized Fokker-Planck-Kolmogorov equation	en
dc.subject.other	Kernel density functions	en
dc.subject.other	Stochastic differential equations	en
dc.subject.other	Partial differential equations	en
dc.title	New partial differential equations governing the joint, response-excitation, probability distributions of nonlinear systems, under general stochastic excitation	en
heal.type	journalArticle	en
heal.identifier.primary	10.1016/j.probengmech.2007.12.028	en
heal.identifier.secondary	http://dx.doi.org/10.1016/j.probengmech.2007.12.028	en
heal.language	English	en
heal.publicationDate	2008	en
heal.abstract	In the present work the problem of determining the probabilistic structure of the dynamical response of nonlinear systems Subjected to general, external, stochastic excitation is considered. The starting point Of Our approach is a Hopf-type equation, governing the evolution of the joint, response-excitation, characteristic functional. Exploiting this equation, we derive new linear partial differential equations governing the joint, response-excitation, characteristic (or probability density) function, which can be considered as an extension of the well-known Fokker-Planck-Kolmogorov equation to the case of a general, correlated excitation and, thus, non-Markovian response character. These new equations are Supplemented by initial conditions and a marginal compatibility condition (with respect to the known probability distribution of the excitation), which is of non-local character. The validity of this new equation is also checked by showing its equivalence with the infinite system of moment equations. The method is applicable to any differential system, in state-space form, exhibiting polynomial nonlinearities. In this paper the method is illustrated through a detailed analysis of a simple, first-order, scalar equation, with a cubic nonlinearity. It is also shown that various versions of Fokker-Planck-Kolmogorov equation, corresponding to the case of independent-increment excitations, can be derived by using the same approach. A numerical method for the solution of these new equations is introduced and illustrated through its application to the simple model problem. It is based on the representation of the joint probability density (or characteristic) function by means of a convex superposition of kernel functions, which permits us to satisfy a priori the non-local marginal compatibility condition. On the basis of this representation, the partial differential equation is eventually transformed to a system of ordinary differential equations for the kernel parameters. Extension to general, multidimensional, dynamical systems exhibiting any polynomial nonlinearity will be presented in a forthcoming paper. (C) 2008 Elsevier Ltd. All rights reserved.	en
heal.publisher	ELSEVIER SCI LTD	en
heal.journalName	Probabilistic Engineering Mechanics	en
dc.identifier.doi	10.1016/j.probengmech.2007.12.028	en
dc.identifier.isi	ISI:000255567300021	en
dc.identifier.volume	23	en
dc.identifier.issue	2-3	en
dc.identifier.spage	289	en
dc.identifier.epage	306	en