A new general law of deformation

Abstract

In this paper a recently developed technique relying on the globally-evolving-based generalized density evolution equation (GE-GDEE) is extended. It is applied to determining the response statistics of multi-dimensional nonlinear systems with fractional derivative elements subject to Gaussian white noise. In particular, the GE-GDEE is derived by a new approach, which enhances its applicability to general continuous processes. Thus, non-Markovian system responses can be treated directly and efficiently. Specifically, for a high-dimensional nonlinear system with fractional derivative elements, of which the responses are non-Markovian, a one- or two-dimensional GE-GDEE, in terms of the response quantity of interest, is obtained. Note that the associated effective drift coefficients involve no fractional terms, and are estimated numerically from data derived from a fairly small number of deterministic analyses. Then, the GE-GDEE is solved by path integration. The accuracy and efficiency of the proposed technique are assessed by applying it for specific nonlinear system and juxtaposing the derived results with pertinent Monte Carlo Simulation data.