Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T02:57:07.868Z Has data issue: false hasContentIssue false
  • English
  • Français

The Approximate Solutions of FPK Equations in High Dimensions for Some Nonlinear Stochastic Dynamic Systems

Published online by Cambridge University Press:  20 August 2015

Guo-Kang Er  and
Vai Pan Iu
Guo-Kang Er*
Affiliation:
Faculty of Science and Technology, University of Macau, Macau SAR, P. R. China
Vai Pan Iu*
Affiliation:
Faculty of Science and Technology, University of Macau, Macau SAR, P. R. China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Get access

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.

Keywords

05.10.Gg05.10.-a02.30.JrNonlinear stochastic dynamic systemslarge-scale systemsprobability density functionFokker-Planck-Kolmogorov equationsubspace
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 5 , November 2011 , pp. 1241 - 1256
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Soong, T. T., Random Differential Equations in Science and Engineering, Academic Press, New York, 1973.Google Scholar
[2]Sobczyk, K., Differential Equations with Application to Physics and Engineering, Kluwer, Boston, 1991.Google Scholar
[3]Socha, L., Linearization in analysis of nonlinear stochastic systems, recent results-Part II: applications, Appl. Mech. Rev., 58 (2005), 303–315.Google Scholar
[4]Lin, Y. K. and Cai, G. Q., Probabilistic Structural Dynamics, McGraw-Hill, New York, 1995.Google Scholar
[5]Scheurkogel, A. and Elishakoff, I., Non-linear random vibration of a two-degree-of-freedom system, in Non-Linear Stochastic Engineering Systems, Eds. Ziegler, F. and Schuëller, G. I., Springer-Verlag, Berlin, 1988, 285–299.Google Scholar
[6]Booton, R. C., Nonlinear control systems with random inputs, IRE Trans. Circuit Theor., 1(1) (1954), 9–19.CrossRefGoogle Scholar
[7]Caughey, T. K., Response of a nonlinear string to random loading, ASME J. Appl. Mech., 26 (1959), 341–344.Google Scholar
[8]Assaf, S. A. and Zirkie, L. D., Approximate analysis of non-linear stochastic systems, Int. J. Control, 23 (1976), 477–492.Google Scholar
[9]Soize, C., Steady-state solution of Fokker-Planck equation in higher dimension, Probab. Eng. Mech., 3 (1988), 196–206.Google Scholar
[10]Tagliani, A., Principle of maximum entropy and probability distributions: definition and applicability field, Probab. Eng. Mech., 4 (1989), 99–104.Google Scholar
[11]Sobczyk, K. and Trebicki, J., Maximum entropy principle in stochastic dynamics, Probab. Eng. Mech., 5 (1990), 102–110.Google Scholar
[12]Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. 1, Gordon and Breach, New York, 1963.Google Scholar
[13]Roberts, J. B. and Spanos, P. D., Stochastic averaging: an approximate method of solving random vibration problems, Int. J. Non-Linear Mech., 21 (1986), 111–134.Google Scholar
[14]Crandall, S. H., Perturbation techniques for random vibration of nonlinear systems, J. Acoust. Soc. Am., 35 (1963), 1700–1705.Google Scholar
[15]Barcilon, V., Singular perturbation analysis of the Fokker-Planck equation: Kramers’ under-damped problem, SIAM J. Appl. Math., 56 (1996), 446–497.Google Scholar
[16]Khasminskii, R. Z. and Yin, G., Asymptotic series for singularly perturbed Kolmogorov-Fokker-Planck equations, SIAM J. Appl. Math., 56 (1996), 1766–1793.CrossRefGoogle Scholar
[17]Lutes, L. D., Approximate technique for treating random vibrations of hysteretic systems, J. Acoust. Soc. Am., 48 (1970), 299–306.CrossRefGoogle Scholar
[18]Cai, G. Q. and Lin, Y. K., A new approximation solution technique for randomly excited nonlinear oscillators, Int. J. Non-Linear Mech., 23 (1988), 409–420.Google Scholar
[19]Zhu, W. Q., Soong, T. T. and Lei, Y., Equivalent nonlinear system method for stochastically excited Hamiltonian systems, ASME J. Appl. Mech., 61 (1994), 618–623.Google Scholar
[20]Spencer, Jr. B. F. and Bergman, L. A., On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems, Int. J. Non-Linear Mech., 4 (1993), 357–372.Google Scholar
[21]Johnson, E. A., Wojtkiewicz, S. F., Bergman, L. A. and Spencer, Jr. B. F., Finite element and finite difference solutions to the transient Fokker-Planck equation, Proceedings of a Workshop: Nonlinear and Stochastic Beam Dynamics in Accelerators, A Challenge to Theoretical and Computational Physics, Lüneburg, Germany, September 29–October 3, 1997, 290–306.Google Scholar
[22]Ujevic, M. and Letelier, P. S., Solving procedure for a 25-diagonal coefficient matrix: direct numerical solutions of the three-dimensional linear Fokker-Planck equation, J. Comput. Phys., 215 (2006), 485–505.Google Scholar
[23]Er, G. K., A consistent method for the solutions to reduced FPK equations in statistical mechanics, Phys. A, 262 (1999), 118–128.Google Scholar
[24]Er, G. K., Zhu, H. T., Iu, V. P. and Kou, K. P., PDF solution of nonlinear oscillators under external and parametric Poisson impulses, AIAA J., 46(11) (2008), 2839–2847.Google Scholar
[25]Er, G. K., The probabilistic solution to non-linear random vibrations of multi-degree-of-freedom systems, ASME J. Appl. Mech., 67 (2000), 355–359.Google Scholar
[26]Harris, C. J., Simulation of multivariate nonlinear stochastic system, Int. J. Numer. Methods Eng., 14 (1979), 37–50.Google Scholar
[27]Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1995.Google Scholar
[28]Er, G. K. and Iu, V. P., A new method for the probabilistic solutions of large-scale nonlinear stochastic dynamic systems, in IUTAM Symposium on Nonlinear Stochastic Dynamics and Control, Eds. Zhu, W. Q., Lin, Y. K. and Cai, G. Q., Springer-Verlag, Berlin, 2011, 25–34.Google Scholar