Abstract The Fokker–Planck–Kolmogorov (FPK) equation, as a well investigated partial differential equation, is of great significance to stochastic dynamics due to its theoretical rigor and exactness. However, practical difficulties with the FPK method are encountered when analysis of multi-degree-of-freedom (MDOF) systems with arbitrary nonlinearity is required. In the present paper, a cell renormalized method (CRM) which is based on a numerical determination of response statistical moments of the FPK equation is developed. Specifically, by invoking the concept of equivalence of probability flux, a cell renormalization procedure and a reconstruction scheme of derivative moments are introduced to divide the continuous state space into a discretized region of cells so that numerical derivative moments is derived. Subsequently, the Cell Renormalized FPK (CR-FPK) equation can be solved by a finite difference algorithm. Two numerical examples are included, and the effectiveness of the proposed method is assessed.

中文翻译：

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随机非线性系统的单元重归一化 FPK 方程

摘要 Fokker-Planck-Kolmogorov (FPK) 方程作为一种经过充分研究的偏微分方程，由于其理论的严谨性和准确性，对随机动力学具有重要意义。然而，当需要分析具有任意非线性的多自由度 (MDOF) 系统时，会遇到 FPK 方法的实际困难。在本文中，开发了一种基于 FPK 方程响应统计矩的数值确定的单元重整化方法 (CRM)。具体而言，通过调用概率通量等价的概念，引入单元重整化过程和导数矩的重构方案，将连续状态空间划分为单元的离散区域，从而导出数值导数矩。随后，Cell Renormalized FPK (CR-FPK) 方程可以通过有限差分算法求解。包括两个数值例子，并评估了所提出方法的有效性。