## Abstract

The finite element method is applied to the solution of the transient Fokker-Planck equation for several often cited nonlinear stochastic systems accurately giving, for the first time, the joint probability density function of the response for a given initial distribution. The method accommodates nonlinearity in both stiffness and damping as well as both additive and multiplicative excitation, although only the former is considered herein. In contrast to the usual approach of directly solving the backward Kolmogorov equation, when appropriate boundary conditions are prescribed, the probability density function associated with the first passage problem can be directly obtained. Standard numerical methods are employed, and results are shown to be highly accurate. Several systems are examined, including linear, Duffing, and Van der Pol oscillators.

Original language | English (US) |
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Pages (from-to) | 357-372 |

Number of pages | 16 |

Journal | Nonlinear Dynamics |

Volume | 4 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1993 |

Externally published | Yes |

## Keywords

- Backward Kolmogorov equation
- Duffing oscillator
- Fokker-Planck equation
- Van der Pol oscillator
- finite elements
- first passage problem
- linear oscillator
- nonlinear vibration
- probability distribution
- random vibration
- stochastic processes

## ASJC Scopus subject areas

- Control and Systems Engineering
- Aerospace Engineering
- Ocean Engineering
- Mechanical Engineering
- Applied Mathematics
- Electrical and Electronic Engineering