Numerical solution of the stationary FPK equation using Shannon wavelets

Journal of Sound and Vibration

Volumn 232, Issue 2, 2000, Pages 405-430

  McWilliam, S ^a   Knappett, D J ^a   Fox, C H J ^a  

^a UNIVERSITY OF NOTTINGHAM   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; FINITE ELEMENT METHOD; FUNCTIONS; LINEAR SYSTEMS; NONLINEAR SYSTEMS; NUMERICAL METHODS; OSCILLATORS (MECHANICAL); PROBABILITY DENSITY FUNCTION; WAVELET TRANSFORMS;

DISTRIBUTED APPROXIMATING FUNCTION; FOKKER-PLANCK-KOLMOGOROV EQUATION; NONLINEAR OSCILLATOR; SHANNON WAVELET;

DYNAMIC RESPONSE;

EID: 0033731439     PISSN: 0022460X     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsvi.1999.2747     Document Type: Article

References (26)

1. Y. K. LIN 1967 Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill.
2. Y. K. LIN and G. Q. CAI 1995 Probabilistic Structural Dynamics: Advanced Theory and Applications. Singapore: McGraw-Hill.
3. J. B. ROBERTS 1982 Journal of Ship Research 26, 229-245. A stochastic theory for nonlinear ship rolling in irregular seas.
4. R. G. BHANDARI and R. E. SHERRER 1968 Journal of Mechanical Engineering Science, 10, 168-174. Random vibrations in discrete non-linear dynamic systems.
5. Y. K. WEN 1975 Proceedings of the ASCE Journal of the Engineering Mechanics Division 101, 389-401. Approximate method for non-linear random vibration.
6. R. S. LANGLEY 1985 Journal of Sound and Vibration 101, 41-54. A finite element method for the statistics of non-linear random vibration.
7. H. P. LANGTANGEN 1991 Reliability Probabilistic Engineering Mechanics 6, 33-48. A general numerical solution method for Fokker-Planck equations with applications to structural.
8. S. F. WOJTKIEWIZC, L. A. BERGMAN and B. F. SPENCER 1994 AAE 94-08, Technical Report, Aeronautical and Astronautical Engineering Department, University of Illinois. Robust numerical solution of the Fokker-Planck-Kolmogorov equation for two dimensional stochastic dynamics systems.
9. J. F. DÜNNE and M. GHANBARI 1997 Journal of Sound and Vibration 206, 697-724. Extreme-value prediction for non-linear oscillators via numerical solutions of the stationary FPK equation.
10. W. YI, S. F. WOJTKIEWIZC, L. A. BERGMAN and B. F. SPENCER 1998 International Conference on Structural Safety and Reliability 1-3, 859-865. Solution of Fokker-Planck equations in higher dimensions: application of the concurrent finite element method.
11. NAESS and J. M. JOHNSEN 1993 Probabilistic Engineering Mechanics 8, 91-106. Response statistics of nonlinear compliant structures by the path integral solution method.
12. A. NAESS 1997 Probabilistic Engineering Mechanics 12, 257-260. IASSAR Report on computational stochastic mechanics.
13. J. S. Yu, G. Q. CAI and Y. K. LIN 1997 International Journal of Non-Linear Mechanics 32, 759-768. A new path integration procedure based on Gauss-Legendre scheme.
14. S. MCWILLIAM 1997 Proceedings of the 6th International Conference on Recent Advances in structural Dynamics, ISVR, Southampton, 1581-1596. Extreme response analysis of non-linear systems to random vibration.
15. G. MUSCOLINO, G. RICCIARDI and M. VASTA 1997 Journal of Non-linear Mechanics 32, 1051-1064. Stationary and non-stationary probability density functions for non-linear oscillators.
16. D. K. HOFFMAN, N. NAYER, O. A. SARAFEDDIN and D. J. KOURI 1991 Journal of Physical Chemistry 95, 8299-8305. Analytic banded approximation for the discretized free propagator.
17. D. K. HOFFMAN, G. W. WEI, D. S. ZHANG and D. J. KOURI 1997 Physical Review E, Part B 56, 1197-1206. Numerical method for the non-linear Fokker-Planck equation.
18. D. S. ZHANG, G. W. WEI, D. J. KOURI and D. K. HOFFMAN 1997 Physics of Fluids 9, 1853-1855. Burgers' equation with high Reynolds number.
19. D. S. ZHANG, G. W. WEI, D. J. KOURI and D. K. HOFFMAN 1998 Chemical Physics Letters 284, 56-62. Lagrange distributed approximating functional method for the solution of the Schrodinger equation.
20. D. K. HOFFMAN, G. W. WEI, D. S. ZHANG and D. J. KOURI 1998 Chemical Physics Letters 287, 119-224. Shannon-Gabor wavelet distributed approximating functional.
21. C. K. CHUI 1992 An Introduction to Wavelets. New York: Academic Press.
22. D. R. HARTREE 1958 Numerical Analysis 93-96. Oxford: Oxford University Press, second Edition.
23. D. E. NEWLAND 1993 An Introduction to Random Vibrations, Spectral and Wavelet Analysis. New York: Longman, third edition.
24. W. F. AMES 1977 Numerical Methods for Partial differential Equations. New York: Academic Press, second edition.
25. W. B. DAVENPORT and W. L. ROOT 1958 An Introduction to the Theory of Random Signals and Noise. New York: McGraw-Hill.
26. S. McWilliam in press Proceedings of the IU T AM Symposium on Non-linearity and Stochastic Structural Dynamics 1999, Madras, India. Numerical solution of the stationary FPK equation for a non-linear oscillator.