一维热传导方程参数识别反问题

论文总字数：31526字

摘 要

一维热传导方程的非齐次项在热传导方程中有着重要的物理意义，它可以被视作长条棒中热传导过程的热源项之一。当非齐次项已知，并且热传导过程的边界条件和初始条件已知时，求解该热传导过程的温度随时间和空间的分布就是热传导方程的正问题。当热传导方程中的非齐次项、热传导率、热传导过程的初始温度等参数未知（部分未知）时，这就需要我们通过热传导过程的某些额外的附加信息来确定热传导过程的上述未知成份，进而求方程的解，从而确定整个热传导过程。这样一类问题就叫做热传导方程的反问题，其本质的困难是问题的不适定性。

本文考虑一维空间上有限长度介质上的热传导模型：

其中, 是已知函数，仅依赖空间变量。

若函数,已知，且满足一定的相容性条件和光滑性，求解上述热传导模型的温度分布称为该过程的正问题,根据已有的偏微分方程理论[15]，该正问题的解是存在唯一的。

本文研究的反问题是当已知，但源项未知时，由末端时刻的测量值

重建。理论上，我们证明了反问题对于精确输入数据解的唯一性，在数值求解上，利用正问题解的级数表达式，将此反问题转化为求解带有正则化罚项的优化问题，利用正则化方法得到反问题的稳定的数值解，并给出数值算例验证了该方法的有效性。

关键词：热传导方程、反问题、不适定性、唯一性、正则化方法、数值解

ABSTRACT

The inhomogeneous term of the heat conduction equation has important significance in the heat conduction equation. It can be regarded as one of the heat source terms of the heat conduction process. When the inhomogeneous term is known and the boundary conditions and initial conditions of the heat conduction process are also known, the determination of the temperature distribution of the heat conduction process is a forward problem of the heat conduction equation. When the inhomogeneous term, heat conductivity and initial temperature of heat conduction process are unknown or partially unknown, we need to determine the above unknown components of heat conduction process by some additional information of heat conduction process, and then find the solution of the equation. Such a kind of problem is called the inverse problem of heat conduction equation.

In this paper, we consider the heat conduction model on a finite length medium in one-dimensional space:

In this model,

are known functions, and the unknown source term f (x) only depends on spatial variables.

If the functions , are known, solving the temperature distribution from the above heat conduction model in one-dimensional space is called forward problem of the process. According to the theory of partial differential equation [15], the solution of the forward problem is unique.

The inverse problem studied in this paper is that, for known ,, we want to determine the unknown source term . We use the measured value at the last moment

to reconstruct f(x). In theory, we prove the uniqueness of this inverse problem. For numerical implementations, based on the series expression of the solution, the inverse problem is transformed into an optimization problem with Tikhonov regularization penalty. The numerical solution of the inverse problem is obtained by using Tikhonov regularization method, and the effectiveness of the method is verified by numerical examples.

Key words: Heat conduction equation, uniqueness, inverse problem, ill-posedness, Tikhonov regularization, numerical solution

目 录

摘要

Abstract

第一章绪论1

1.1热传导反问题的背景1

1.2热传导反问题已有的工作1

1.3本文的主要工作2

第二章预备知识3

2.1一般正则化理论3

2.2 正则化方法5

第三章热传导方程正问题6

3.1求解表达式6

3.2 差分格式8

3.3数值求解10

第四章热传导方程反问题15

4.1 热传导方程反问题解的唯一性15

4.2 反问题解数值求解15

4.3 研究不同的对反演结果的影响21

第五章结束语25

致谢26

参考文献27

第一章 绪论

1.1热传导反问题的背景

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