兰州大学机构库 >数学与统计学院
几类抛物型方程逆问题的数值方法研究
Alternative TitleStudies on Numerical Methods for Several Inverse Problems for Parabolic Equations
温瑾
Thesis Advisor魏婷
2011-06-01
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword不适定问题 反向热传导问题 反热源问题 基本解方法 数值微分 正则化方法 广义交叉核实 L-曲线
Abstract本文考虑了抛物型方程的几类逆问题, 包括同时反演热源和初值的热传导反问题和带有积分条件的二维抛物型方程边界系数辨识问题. 其中, 第一类反演初值的问题包括反演部分初值及全部初值, 而且它们所带的边界条件也有区别, 一种是Dirichlet条件, 另一种是Neumann条件. 首先, 我们给出了反演热源和部分初值的热传导反问题, 通过对未知热源函数作积分变换, 将原问题化为齐次问题, 将反热源问题化成一个反边界值问题.因此, 我们可以用基本解方法来考虑. 其次, 我们主要研究了反演热源和全部初值的热传导反问题, 通过函数变换的技巧, 我们给出了唯一性定理的一种证明. 同时, 利用这一变换, 我们也将该问题化成一个反向热传导问题和一个反热源问题. 对于反向热传导问题, 我们用基本解方法来求解. 而在反演热源的过程中, 我们还用到了数值微分的技巧. 最后, 我们研究了一类积分条件下的二维抛物型方程反边界系数问题, 由于方程本身是齐次的, 所以我们采用二维基本解方法来求解. 因为基本解方法得到的系数矩阵是高度病态的, 所以我们采用了Tikhonov正则化方法来求解. 关于正则化参数的选取, 我们利用两种后验选取方式, 即广义交叉核实准则和L-曲线方法. 而在求解数值微分的过程中, 正则化参数是先验选取的. 对于以上的几类问题, 我们均给出了若干典型的数值例子来验证我们方法的稳定性和有效性. 从数值结果可以看出, 本文的方法的确能够很好地解决我们提出的问题. 此外, 我们还通过数据分析, 对本文的一些结论进行了总结.
Other AbstractIn this thesis, we consider several classes of inverse problems for parabolic equations, including simultaneous reconstruction of the heat source and the initial temperature, and determining the boundary coefficient of two-dimensional parabolic equation with the integral condition. There are two kinds of reconstructions of initial temperature in the first class, i.e., the partial and entire initial values. Besides, the boundary conditions are different from each other; one is under Dirichlet conditions, and the other is under Neumann conditions. Firstly, we give the inverse heat conduction problem of simultaneous reconstruction of the heat source and the partial initial temperature. By using an integral transformation of the unknown heat source function, we change the former problem into a homogeneous one, and transform the inverse source problem into an inverse boundary value problem. Hence, we can use the method of fundamental solutions. Secondly, we mainly investigate the problem of reconstructing the heat source and the entire initial value. We give a proof of the uniqueness theorem of our proposed problem by applying a function transformation. Meanwhile, we also change the problem into a backward heat conduction problem and an inverse source problem. For the backward heat conduction problem, the method of fundamental solutions is applied to solve it. In the process of solving the heat source, we use the trick of the numerical differentiation. Finally, we study the inverse boundary coefficient problem of two-dimensional parabolic equation with the integral condition. Because of the homogeneity of the equation, we can use the two-dimensional fundamental solution method. Since the resultant matrices of the method of fundamental solutions are highly ill-conditioned, we utilize the Tikhonov regularization method to solve them. For the choices of the regularization parameters, we use two kinds of a-posteriori schemes, i.e., the generalized cross-validation and L-curve methods. Besides, we choose the regularization parameters by a-priori scheme in the numerical differentiation. For the above problems, we give several typical numerical examples to show the stability and efficiency of our proposed methods. From the numerical results, we can obtain the conclusion that our methods work effectively on these ill-posed problems. Moreover, we also summarize some conclusions about our methods through data analysis.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225038
Collection数学与统计学院
Recommended Citation
GB/T 7714
温瑾. 几类抛物型方程逆问题的数值方法研究[D]. 兰州. 兰州大学,2011.
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