Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques

We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic exci...

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Bibliographic Details
Published inMathematics (Basel) Vol. 9; no. 3; p. 204
Main Authors Cortés, Juan-Carlos, López-Navarro, Elena, Romero, José-Vicente, Roselló, María-Dolores
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.01.2021
Subjects
Approximation
Differential equations
Earthquakes
Excitation
Food science
Gaussian process
Maximum entropy
maximum entropy principle
Noise
Nonlinearity
Oscillators
Partial differential equations
Perturbation methods
probability density function
Probability density functions
random nonlinear oscillator
Random vibration
stationary Gaussian noise
Stochastic models
stochastic perturbations
Stochastic processes
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Summary:We combine the stochastic perturbation method with the maximum entropy principle to construct approximations of the first probability density function of the steady-state solution of a class of nonlinear oscillators subject to small perturbations in the nonlinear term and driven by a stochastic excitation. The nonlinearity depends both upon position and velocity, and the excitation is given by a stationary Gaussian stochastic process with certain additional properties. Furthermore, we approximate higher-order moments, the variance, and the correlation functions of the solution. The theoretical findings are illustrated via some numerical experiments that confirm that our approximations are reliable.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9030204