ISSN:
1573-269X
Keywords:
Backward Kolmogorov equation
;
Duffing oscillator
;
finite elements
;
first passage problem
;
Fokker-Planck equation
;
linear oscillator
;
nonlinear vibration
;
probability distribution
;
random vibration
;
stochastic processes
;
Van der Pol oscillator
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00120671
