Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

A novel approximate analytical technique for determining the non-stationary response probability density function (PDF) of randomly excited linear and nonlinear oscillators endowed with fractional derivatives elements is developed. Specifically, the concept of the Wiener path integral in conjunction with a variational formulation is utilized to derive an approximate closed form solution for the system response non-stationary PDF. Notably, the determination of the non-stationary response PDF is accomplished without the need to advance the solution in short time steps as it is required by the existing alternative numerical path integral solution schemes which rely on a discrete version of the Chapman–Kolmogorov (C–K) equation. This is accomplished by circumventing the solution of the associated Euler–Lagrange equation ordinarily used in the path integral based procedures. The accuracy of the technique is demonstrated by pertinent Monte Carlo simulations.