Lévy噪声下非线性随机动力系统统计特性的卷积求积方法
Statistics of non-linear stochastic dynamical systems under Lévy noises by a convolution quadrature approach
摘要：
本文提出了一种新的数值方法，用于求解标量非线性系统在Lévy白噪声激励下的非平稳响应的统计特性。所提出的数值方法依赖于引入维纳-霍普夫型的积分变换到控制特征函数的方程。一旦这个方程被重写为偏积分微分方程，它然后通过应用卷积求积的方法最初提出的卢比希，在这里扩展到处理这个特定的积分变换来解决。所提出的方法是相关的，有两个原因：1）具有几个不同的漂移项的系统的统计可以被处理在一个有效的方式，独立于白噪声的种类; 2）维纳-霍普夫积分变换的特殊形式和它的数值计算，都在这项研究中介绍，是广义的分数积分微分算子的潜在类型和Grünwald-Letnikov分数导数，分别。
英文摘要：
This paper describes a novel numerical approach to find the statistics of the non-stationary response of scalar non-linear systems excited by Lévy white noises. The proposed numerical procedure relies on the introduction of an integral transform of Wiener-Hopf type into the equation governing the characteristic function. Once this equation is rewritten as partial integro-differential equation, it is then solved by applying the method of convolution quadrature originally proposed by Lubich, here extended to deal with this particular integral transform. The proposed approach is relevant for two reasons: 1) Statistics of systems with several different drift terms can be handled in an efficient way, independently from the kind of white noise; 2) The particular form of Wiener-Hopf integral transform and its numerical evaluation, both introduced in this study, are generalizations of fractional integro-differential operators of potential type and Grünwald-Letnikov fractional derivatives, respectively.
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