Solution to the Backward-Kolmogorov Equation for a Nonstationary Oscillation Problem

1982 ◽  
Vol 49 (4) ◽  
pp. 923-925 ◽  
Author(s):  
G. P. Solomos ◽  
P-T. D. Spanos
Keyword(s):  
Random Excitation ◽  
Response Amplitude ◽  
Kolmogorov Equation ◽  
Linear Oscillator ◽  
Markov Approximation ◽  
Nonstationary Oscillation ◽  
Oscillation Problem ◽  
Backward Kolmogorov Equation

The solution of a backward-Kolmogorov equation is presented. This equation is associated with a Markov approximation of the response amplitude of a lightly damped linear oscillator driven by an evolutionary random excitation.

scholarly journals Spectral analysis of vibration in weakly non-linear systems

2000 ◽  
Vol 22 (3) ◽  
pp. 181-192
Author(s):  
Nguyen Tien Khiem
Keyword(s):  
Nonlinear Systems ◽  
Spectral Density ◽  
Density Function ◽  
Asymptotic Methods ◽  
Random Excitation ◽  
Spectral Density Function ◽  
Kolmogorov Equation ◽  
Spectral Structure ◽  
Random Load ◽  
Weakly Nonlinear

The weakly nonlinear systems subjected to deterministic excitations have been fully and deeply studied by use of the well developed asymptotic methods [1-4]. The systems excited by a random load have been investigated mostly using the Fokker-Plank-Kolmogorov equation technique combined with the asymptotic methods [5-8]. However, the last approach in most successful cases allows to obtain only a stationary single point probability density function, that contains no information about the correlation nor' consequently, the spectral structure of the response. The linearization technique [9, 10] in general permits the spectral density of the response to be determined, but the spectral function obtained by this method because of the linearization eliminates the effect of the nonlinearity. Thus, spectral structure of response of weakly nonlinear systems to random excitation, to the author's knowledge, has not been studied enough. This paper deals with the above mentioned problem. The main idea of this work is the use of an analytical simulation of random excitation given by its spectral density function and afterward application of the well known procedure of the asymptotic method to obtain an asymptotic expression of the response spectral density function. The obtained spectral relationship covers the linear system case and especially emphasizes the nonlinear effect on the spectral density of response. The theory will be illustrated by an example and at the end of this paper there will be a discussion about the obtained results.  

scholarly journals The nonlocal boundary value problem for one-dimensional backward Kolmogorov equation and associated semigroup

2019 ◽  
Vol 11 (2) ◽  
pp. 463-474
Author(s):  
R.V. Shevchuk ◽  
I.Ya. Savka ◽  
Z.M. Nytrebych
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Boundary Integral ◽  
Kolmogorov Equation ◽  
Linear Parabolic Equation ◽  
One Dimensional ◽  
Wentzell Boundary Condition ◽  
Backward Kolmogorov Equation

This paper is devoted to a partial differential equation approach to the problem of construction of Feller semigroups associated with one-dimensional diffusion processes with boundary conditions in theory of stochastic processes. In this paper we investigate the boundary-value problem for a one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) in curvilinear bounded domain with one of the variants of nonlocal Feller-Wentzell boundary condition. We restrict our attention to the case when the boundary condition has only one term and it is of the integral type. The classical solution of the last problem is obtained by the boundary integral equation method with the use of the fundamental solution of backward Kolmogorov equation and the associated parabolic potentials. This solution is used to construct the Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle leaves the boundary of the domain by jumps.

scholarly journals On Deterministic and Stochastic Multiple Pathogen Epidemic Models

Epidemiologia ◽  
2021 ◽  
Vol 2 (3) ◽  
pp. 325-337
Author(s):  
Fernando Vadillo
Keyword(s):  
Variable Coefficients ◽  
Epidemic Models ◽  
Kolmogorov Equation ◽  
The Finite Element Method ◽  
Persistence Time ◽  
Main Challenge ◽  
Stochastic Epidemic ◽  
The Mean ◽  
Backward Kolmogorov Equation ◽  
Multiple Pathogen

In this paper, we consider a stochastic epidemic model with two pathogens. In order to analyze the coexistence of two pathogens, we compute numerically the expectation time until extinction (the mean persistence time), which satisfies a stationary partial differential equation with degenerate variable coefficients, related to backward Kolmogorov equation. I use the finite element method in order to solve this equation, and we implement it in FreeFem++. The main conclusion of this paper is that the deterministic and stochastic epidemic models differ considerably in predicting coexistence of the two diseases and in the extinction outcome of one of them. Now, the main challenge would be to find an explanation for this result.

Stochastic linearization: the theory

1998 ◽  
Vol 35 (3) ◽  
pp. 718-730 ◽  
Author(s):  
Pierre Bernard ◽  
Liming Wu
Keyword(s):  
Large Deviation ◽  
Random Excitation ◽  
Kolmogorov Equation ◽  
Parabolic Partial Differential Equations ◽  
Statistical Linearization ◽  
Linearization Methods ◽  
Large Deviation Principles ◽  
Kullback Information ◽  
Stochastic Linearization ◽  
Global Linearization

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.

Stochastic linearization: the theory

1998 ◽  
Vol 35 (03) ◽  
pp. 718-730 ◽  
Author(s):  
Pierre Bernard ◽  
Liming Wu
Keyword(s):  
Large Deviation ◽  
Random Excitation ◽  
Kolmogorov Equation ◽  
Parabolic Partial Differential Equations ◽  
Statistical Linearization ◽  
Linearization Methods ◽  
Large Deviation Principles ◽  
Kullback Information ◽  
Stochastic Linearization ◽  
Global Linearization

Very little is known about the quantitative behaviour of dynamical systems with random excitation, unless the system is linear. Known techniques imply the resolution of parabolic partial differential equations (Fokker–Planck–Kolmogorov equation), which are degenerate and of high dimension and for which there is no effective known method of resolution. Therefore, users (physicists, mechanical engineers) concerned with such systems have had to design global linearization techniques, known as equivalent statistical linearization (Roberts and Spanos (1990)). So far, there has been no rigorous justification of these techniques, with the notable exception of the paper by Kozin (1987). In this contribution, using large deviation principles, several mathematically founded linearization methods are proposed. These principles use relative entropy, or Kullback information, of two probability measures, and Donsker–Varadhan entropy of a Gaussian measure relatively to a Markov kernel. The method of ‘true linearization’ (Roberts and Spanos (1990)) is justified.

DISPERSIVE GROUNDWATER FLOW AND POLLUTION

1991 ◽  
Vol 01 (01) ◽  
pp. 61-81 ◽  
Author(s):  
ONNO A. VAN HERWAARDEN ◽  
JOHAN GRASMAN
Keyword(s):  
Random Walk ◽  
Dirichlet Problem ◽  
Groundwater Flow ◽  
Arrival Time ◽  
Kolmogorov Equation ◽  
Dispersive Flow ◽  
Backward Kolmogorov Equation ◽  
Protected Zone

By solving asymptotically the Dirichlet problem for the backward Kolmogorov equation describing the random walk of a particle in a dispersive flow, it is computed at what rate contaminated particles cross the boundary of a protected zone. The method also yields an estimate of the expected arrival time.

scholarly journals On backward Kolmogorov equation related to CIR process

2018 ◽  
Vol 5 (1) ◽  
pp. 113-127 ◽  
Author(s):  
Vigirdas Mackevičius ◽  
Gabrielė Mongirdaitė
Keyword(s):  
Kolmogorov Equation ◽  
Backward Kolmogorov Equation ◽  
Cir Process

Probability of First-Passage Failure for Nonstationary Random Vibration

1975 ◽  
Vol 42 (3) ◽  
pp. 716-720 ◽  
Author(s):  
J. B. Roberts
Keyword(s):  
White Noise ◽  
Barrier Height ◽  
Random Vibration ◽  
Series Solution ◽  
Short Pulse ◽  
Random Excitation ◽  
Numerical Results ◽  
Linear Oscillator ◽  
First Passage ◽  
First Passage Failure

The problem of calculating the probability of first-passage failure, Pf, is considered, for systems responding to a short pulse of nonstationary random excitation. It is shown, by an analysis based on the “in and exclusion” series, that, under certain conditions, Pf tends to Pf*, the probability calculated by assuming Poisson distributed barrier crossings, as the barrier height, b, tends to infinity. The first three terms in the series solution provide bounds to Pf which converge when b is large. Methods of estimating Pf from these terms are presented which are useful even when the series is divergent. The theory is illustrated by numerical results relating to a linear oscillator excited by modulated white noise.

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