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一类带有空间时间白噪音随机弹性方程的全离散差分格式

张英晗, 杨小远   

  1. 北京航空航天大学数学与系统科学学院, 北京 100191
  • 收稿日期:2014-10-13 出版日期:2016-02-15 发布日期:2016-01-22
  • 基金资助:

    国家自然科学基金(61271010);北京市自然科学基金(4152029)及北京航空航天大学博士创新基金资助项目

张英晗, 杨小远. 一类带有空间时间白噪音随机弹性方程的全离散差分格式[J]. 计算数学, 2016, 38(1): 25-46.

Zhang Yinghan, Yang Xiaoyuan. FULL-DISCRETIZATION DIFFERENCE SCHEME FOR A STOCHASTIC ELASTIC EQUATION DRIVEN BY SPACE-TIME WHITE NOISE[J]. Mathematica Numerica Sinica, 2016, 38(1): 25-46.

FULL-DISCRETIZATION DIFFERENCE SCHEME FOR A STOCHASTIC ELASTIC EQUATION DRIVEN BY SPACE-TIME WHITE NOISE

Zhang Yinghan, Yang Xiaoyuan   

  1. Department of Mathematics, Beihang University, LMIB of the Ministry of Education, Beijing 100191, China
  • Received:2014-10-13 Online:2016-02-15 Published:2016-01-22
随机弹性方程在结构工程中有许多应用.本文研究一类由空间时间白噪音扰动的随机弹性方程的全离散有限差分格式.通过引入新的函数,将随机弹性方程表示成一阶方程组的形式,然后对噪音项进行分片常数逼近,构造了带有空间时间白噪音随机弹性方程的全离散差分格式.基于对Gronwall不等式和Burkholder不等式的应用,证明了格式的Lp收敛性并得到了收敛阶.在数值实验中结合Monte-Carlo方法,所得实验结果与理论分析是一致的.
Stochastic elastic equations have many applications in structural engineering. The fully discrete finite difference scheme for a class of stochastic elastic equations driven by space-time white noise are studied in this paper. By introducing a new function, the stochastic elastic equation is expressed as a first-order system of equations, then a piecewise constant approximation of the noise term is conducted, a full-dicretization difference scheme for a nonlinear stochastic elastic equation driven by a space-time white noise is obtained. Based on Burkholder's inequality and Gronwall's inequality, we prove Lp-convergence of the scheme and determine the rate of convergence. Almost sure convergence, uniformly in time and space, is also obtained. By using Monte-Carlo method, the experimental results are consistent with the theoretical analysis.

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