Transient probabilistic solutions of stochastic oscillator with even nonlinearities by exponential polynomial closure method

The current work is devoted to analyze the transient probability density function solutions of stochastic oscillator with even nonlinearities under external excitation of Gaussian white noise by applying the extended exponential polynomial closure method. Specifically, the Fokker–Planck–Kolmogorov equation which governs the probability density function solutions of the nonlinear system is presented first. The residual error of the Fokker–Planck–Kolmogorov equation is then derived by assuming the probability density function solution as the type of exponential polynomial with time-dependent variables. Finally, by making the projection of the residual error vanish, a set of nonlinear ordinary differential equations is established and solved numerically. Numerical analysis show that the extended exponential polynomial closure method with polynomial order being six is both effective and efficient for solving the transient analysis of the stochastic oscillator with even nonlinearities by comparing the numerical results obtained by the proposed method with those obtained by Monte Carlo simulation method. Numerical results also show that the transient probability density function solutions of the system responses are not symmetric about their nonzero means due to the existence of even nonlinearities.