Memory-Dependent Fokker–Planck–Kolmogorov Equations for Nonlinear SDOF Oscillators Under Fractional Gaussian Noise Excitation

This paper investigates the transient probabilistic responses of nonlinear single-degree-of-freedom oscillators subjected to external fractional Gaussian noise (FGN) excitation. Owing to the inherent long-range correlations and memory characteristics of FGN, the resulting response process exhibits non-Markovian properties, rendering the classical Fokker–Planck–Kolmogorov (FPK) equation method inapplicable in its direct form. To overcome this critical challenge, a memory-dependent FPK (memFPK) equation is formulated for two-dimensional nonlinear stochastic systems within the fractional Wick–Itô–Skorohod integral framework. The derived memFPK equation incorporates mixed second-order derivative terms, as well as time-dependent and state-dependent diffusion coefficients, which inherently capture the long-range correlations and memory effects induced by FGN excitation. For the numerical solution of the memFPK equation, a discretized local mean treatment is developed to estimate the memory-dependent diffusion coefficients involving conditional expectations. The proposed approach integrates local statistical averaging and smoothing techniques to enhance the stability of coefficient estimation. Subsequently, the memFPK equation is numerically solved using a finite difference scheme. The accuracy and effectiveness of the proposed framework are validated through linear and nonlinear numerical examples. Comparative results demonstrate the excellent agreement with analytical solutions or Monte Carlo simulations in terms of transient joint probability density functions (PDFs), marginal PDFs, low-probability tail regions, and statistical moments. These findings confirm that the proposed memFPK equation method serves as a robust and effective tool for analyzing the transient non-Markovian probabilistic responses of nonlinear single-degree-of-freedom (SDOF) systems under FGN excitation.