| 蒙特卡罗方法在解微分方程边值问题中的应用 |
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| 引用本文： | 左应红,王建国.蒙特卡罗方法在解微分方程边值问题中的应用[J].强激光与粒子束,2012,24(12):3023-3027. |
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| 作者姓名： | 左应红 王建国 |
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| 作者单位： | 1.西北核技术研究所, 西安 71 0024; |
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| 摘 要： | 介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想，结合有限差分方法，建立了求解微分方程边值问题的随机概率模型，并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例，研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明:利用蒙特卡罗方法，不仅可以有效解决给定边界条件的微分方程，对于给定初值条件的微分方程，也可以从时域有限差分方程出发，采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明，增加蒙特卡罗试验中的模拟粒子点数，可以提高计算结果的精度。
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| 关 键 词： | 蒙特卡罗方法 微分方程 边值问题 随机概率模型 热传导方程 |
| 收稿时间： | 2012/7/2 |
Application of Monte Carlo method to solving boundary value problem of differential equations |
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| Affiliation: | 1.Northwest Institute of Nuclear Technology,P.O.Box 69-12,Xi’an 710024,China;2.Department of Engineering Physics,Tsinghua University,Beijing 100084,China;3.School of Electronic and Information Engineering,Xi’an Jiaotong University,Xi’an 710049,China |
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| Abstract: | This paper introduces the foundation of the Monte Carlo method and the way how to generate the random numbers. Based on the basic thought of the Monte Carlo method and the finite differential method, the stochastic model for solving the boundary value problem of differential equations is built. To investigate the application of the Monte Carlo method to solving the boundary value problem of differential equations, the model is used to solve Laplace’s equations with the first boundary condition and the unsteady heat transfer equation with initial values and boundary conditions. The results show that the boundary value problem of differential equations can be effectively solved with the Monte Carlo method, and the differential equations with initial condition can also be calculated by using a stochastic probability model which is based on the time-domain finite differential equations. Both the simulation results and theoretical analyses show that the errors of numerical results are lowered as the number of simulation particles is increased. |
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| Keywords: | Monte Carlo method differential equations boundary value problem stochastic probability model heat transfer equation |
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