On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems

B. F. Spencer, L. A. Bergman

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)357-372
Number of pages16
JournalNonlinear Dynamics
Issue number4
StatePublished - Aug 1993


  • 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

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


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