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Beyond the hypothesis of boundedness for the random coefficient of the Legendre differential equation with uncertainties
Applied Mathematics and Computation ( IF 3.5 ) Pub Date : 2021-02-01 , DOI: 10.1016/j.amc.2020.125638
Marc Jornet

Abstract In this paper, we aim at relaxing the boundedness condition for the input coefficient A of the Legendre random differential equation, to permit important unbounded probability distributions for A. We demonstrate that the formal solution constructed using the Frobenius approach is indeed the mean square solution on the domain ( − 1 , 1 ) , under mean fourth integrability of the initial conditions X0, X1 and sublinear growth of the 8n-th norm of A. Under linear growth of the 8n-th norm of A, the mean square solution is only defined on a neighborhood of zero contained in ( − 1 , 1 ) . These conditions are closely related to the finiteness of the moment-generating function of A. Numerical experiments on the approximation of the solution statistics for unbounded equation coefficients A illustrate the theoretical findings.

中文翻译:

具有不确定性的勒让德微分方程随机系数的有界假设

摘要 在本文中,我们旨在放宽勒让德随机微分方程的输入系数 A 的有界条件,以允许 A 的重要无界概率分布。 我们证明使用 Frobenius 方法构造的形式解确实是均方解在域 ( − 1 , 1 ) 上,在初始条件 X0、X1 的平均四次可积性和 A 的第 8n 范数的次线性增长下。 在 A 的第 8n 范数的线性增长下,均方解为仅在 (−1, 1) 中包含的零邻域上定义。这些条件与 A 的矩生成函数的有限性密切相关。对无界方程系数 A 的解统计近似值的数值实验说明了理论发现。
更新日期:2021-02-01
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