An approximate analytical technique is developed for determining, in closed form, the transition probability density function (PDF) of a general class of first-order stochastic differential equations (SDEs) with nonlinearities both in the drift and in the diffusion coefficients. Specifically, first, resorting to the Wiener path integral most probable path approximation and utilizing the Cauchy–Schwarz inequality yields a closed-form expression for the system response PDF, at practically zero computational cost. Next, the accuracy of this approximation is enhanced by proposing a more general PDF form with additional parameters to be determined. This is done by relying on the associated Fokker–Planck operator to formulate and solve an error minimization problem. Besides the mathematical merit of the derived closed-form approximate PDFs, an additional significant advantage of the technique relates to the fact that it can be readily coupled with a stochastic averaging treatment of second-order SDEs governing the dynamics of diverse stochastically excited nonlinear/hysteretic oscillators. In this regard, it is shown that the technique is capable of determining approximately the response amplitude transition PDF of a wide range of nonlinear oscillators, including hysteretic systems following the Preisach versatile modeling. Several numerical examples are considered for demonstrating the reliability and computational efficiency of the technique. Comparisons with pertinent Monte Carlo simulation data are provided as well.
- Cauchy–Schwarz inequality
- Fokker–Planck equation
- Nonlinear stochastic dynamics
- Path integral
- Stochastic differential equations