^{a}, Z.G. Kapelonis

^{b}and K.I. Mamis

^{c}

^{a}mathan@central.ntua.gr

^{b}zkapel@central.ntua.gr

^{c}konmamis@central.ntua.gr

Solving a nonlinear random differential equation (RDE) excited by Gaussian colored noise is an important problem arising in both physics and engineering, whose difficulty lies in the non-Markovian nature of its response. In the present work, approximate generalized Fokker-Planck-Kolmogorov (genFPK) equations that govern the evolution of the probability density function of the response are formulated. These genFPK equations are derived from the stochastic Liouville equation corresponding to the RDE by a novel systematic approximation of the non-local terms, which reflect the non-Markovianity of the response. This approximation is implemented by using an extension of the classical Novikov-Furutsu theorem, and utilizing the Volterra-Taylor expansion to perform an efficient closure. To the best of our knowledge, this treatment has not been presented before. The family of genFPK equations obtained contain, as special cases, existing results, and reproduce and generalize in a rational way Hänggi’s ansatz. Particular forms of the presented genFPK equations are studied numerically in the transient regime, by implementing a numerical scheme based on the partition of unity finite element method (PUFEM). The direct numerical solutions are compared to Monte Carlo simulation results, for the comparative assessment of the genFPK equations. Although, herein, a scalar RDE is considered, everything can be extended in a straightforward (yet laborious) way to systems of nonlinear RDEs.