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

Author

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  • Juan-Carlos Cortés

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
    These authors contributed equally to this work.)

  • Elena López-Navarro

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
    These authors contributed equally to this work.)

  • José-Vicente Romero

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
    These authors contributed equally to this work.)

  • María-Dolores Roselló

    (Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
    These authors contributed equally to this work.)

Abstract

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.

Suggested Citation

  • Juan-Carlos Cortés & Elena López-Navarro & José-Vicente Romero & María-Dolores Roselló, 2021. "Approximating the Density of Random Differential Equations with Weak Nonlinearities via Perturbation Techniques," Mathematics, MDPI, vol. 9(3), pages 1-17, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:204-:d:483532
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    References listed on IDEAS

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    1. Gitterman, M., 2014. "Stochastic oscillator with random mass: New type of Brownian motion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 11-21.
    2. de la Cruz, H., 2020. "Stabilized explicit methods for the approximation of stochastic systems driven by small additive noises," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    3. Yasir Khan & Hector Vázquez-Leal & Luis Hernandez-Martínez, 2012. "Removal of Noise Oscillation Term Appearing in the Nonlinear Equation Solution," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-9, August.
    4. Calatayud, J. & Cortés, J.-C. & Jornet, M., 2018. "The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 261-279.
    5. Mohamed A. El-Beltagy & Amnah S. Al-Johani, 2013. "Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-21, August.
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    Cited by:

    1. Lucas Jódar & Rafael Company, 2022. "Preface to “Mathematical Methods, Modelling and Applications”," Mathematics, MDPI, vol. 10(9), pages 1-2, May.

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