具有解析不确定性的二阶线性微分方程：通过计算密度函数的随机分析
Second order linear differential equations with analytic uncertainties: stochastic analysis via the computation of the probability density function
摘要：
本文研究随机二阶线性微分方程的分析问题。通常，求解这些方程包括计算响应过程的第一个统计量，并且该任务一直是文献中的基本目标。一个更雄心勃勃的目标是计算的解决方案密度函数。本文在一般随机非自治二阶线性微分方程的解析数据处理的情形下，给出了这两方面的进展。采用Fröbenius方法得到了均方收敛的幂级数形式的随机解。我们证明了收敛性要求随机输入系数的有界性。此外，Fröbenius方法的均方误差被证明是指数下降的级数中的条款的数量，虽然不是均匀的时间。对于给定时刻解的密度函数，我们依赖于全概率定律将其以封闭形式表示为期望。为了计算这个期望值，通过使用基本集的截断幂级数来降低问题的维数，构造了一系列近似密度函数。证明了关于密度函数序列的逐点收敛性和全变差收敛性的几个理论结果。在Lipschitz假设下，点态收敛是指数收敛的。由于密度函数是用期望值表示的，我们提出了一种符号蒙特卡罗抽样算法来估计密度函数。该算法的实现和应用的几个数值例子，旨在说明本文的理论研究结果。
英文摘要：
This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random non-autonomous second order linear differential equations with analytic data processes. The Fröbenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Fröbenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, we rely on the law of total probability to express it in closed-form as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a symbolic Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
