The closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for nonlinear systems

• Develop a new method for finding steady-state PDFs of FPK.
• Improve the method of weighted residue with iterations.
• Study several difficult examples of nonlinear stochastic systems.
• Examine the property of exponential polynomial.

In this paper, we propose a new method to obtain the closed-form solution of the reduced Fokker–Planck–Kolmogorov equation for single degree of freedom nonlinear systems under external and parametric Gaussian white noise excitations. The assumed stationary probability density function consists of an exponential polynomial with a logarithmic term to account for parametric excitations. The undetermined coefficients in the assumed solution are computed with the help of an iterative method of weighted residue. We have found that the iterative process generates a sequence of solutions that converge to the exact solutions if they exist. Three examples with known exact steady-state probability density functions are used to demonstrate the convergence of the proposed method.