识别热方程移动边界的一种基本解方法 Alternative Title The Method of Fundamental Solutions for Detection of a Moving Boundary in Heat Equation 徐州 Subtype 硕士 Thesis Advisor 魏婷 2012-05-30 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 热方程 不适定性 基本解方法 移动边界 Abstract 本文采用一种基本解方法, 对一个热方程上的移动边界识别问题进行求解. 该问题只提供了部分边界上面的Cauchy数据, 是一个严重不适定的问题. 本文对该问题求解的主要过程是: 首先, 根据基本解方法的基本思想, 用热方程基本解的一个线性组合近似逼近原问题的解, 而且这些基本解的资源点都选取在求解区域之外. 然后, 利用原问题的边界条件, 将问题转化为一个无约束最优化问题, 即极小化一个目标泛函, 而线性组合中的未知系数以及未知移动边界的信息就作为该目标泛函中的未知量. 由于该问题的不适定性, 即输入的边界上的Cauchy数据发生一个微小的变化, 就会导致所得到的解(未知移动边界)发生一个较大的变化, 因此, 为了在输入的边界数据存在误差的时候得到稳定的数值解, 我们先对目标泛函进行一种修正, 调节目标泛函中不同残差项之间的权重. 随后, 我们对修正后的目标泛函进行离散化, 使其变成一个多元函数, 该多元函数是一些非线性多元函数平方和的形式. 最后, 我们利用数值软件MATLAB对进行了修正后的离散形式的目标函数求极小值点, 未知移动边界的数值解就可以从所求出的极小值点中得到了. Other Abstract In this paper, the method of fundamental solutions(MFS) is used to determine a moving boundary from Cauchy data in one dimensional heat equation. This problem is severely ill-posed. The main process of solving this problem is follows: First, according to the main idea of the MFS, we approximate an unknown solution of the heat equation by a linear combination of fundamental solutions whose singularities are located outside the solution domain. Second, the coefficients and the unknown moving boundary will be determined by imposing the boundary conditions in an unconstrained optimization problem, i.e. minimize an objective function. Because of the ill-posedness of this problem, i.e. any small change on the input Cauchy data on boundary can result in a dramatic change to the solution(unknown moving boundary), in order to obtain a stable solution when the Cauchy data is disturbed, we will amend the objective function for adjusting the weight of residual terms in the objective function. Then, we discrete the objective function amended into a sum of squares of nonlinear multivariate function. Finally, we will use numerical software MATLAB to solve the unconstrained optimization problem and obtain the minimizer which contains the numerical solution of the moving boundary. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224703 Collection 数学与统计学院 Recommended CitationGB/T 7714 徐州. 识别热方程移动边界的一种基本解方法[D]. 兰州. 兰州大学,2012.
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