This paper presents a wavelet-based method for determining response evolutionary power spectrum density (EPSD) of linear/non-linear systems endowed with fractional derivative elements. Specifically, first, the generalized harmonic wavelets (GHWs) based Galerkin approximation of the stochastic processes are utilized to transform the fractional-order stochastic linear/non-linear differential equations into a set of linear/non-linear algebraic equations with unknown response wavelet coefficients. Next, the linear algebraic equations are solved in a closed-form, while the non-linear ones are treated by the gradient-based standard numerical methods. Further, an analytical relationship between the EPSD of the excitation and of the response for a linear system is derived by considering the wavelet representation of stochastic processes. For a non-linear system, the response EPSD is estimated by repeated solving of sample algebraic equations. Pertinent numerical examples demonstrate the applicability and accuracy of the proposed method.

本文提出了一种基于小波的方法来确定具有分数阶导数元素的线性/非线性系统的响应进化功率谱密度（EPSD）。具体而言，首先，利用基于广义谐波小波（GHW）的随机过程的伽辽金近似，将分数阶随机线性/非线性微分方程转化为一组响应小波系数未知的线性/非线性代数方程。接下来，线性代数方程以封闭形式求解，而非线性方程则通过基于梯度的标准数值方法处理。此外，通过考虑随机过程的小波表示，推导出线性系统的激励和响应的 EPSD 之间的解析关系。对于非线性系统，通过重复求解样本代数方程来估算响应EPSD。相关的数值例子证明了所提出方法的适用性和准确性。