一类热传导方程初边值问题解的适定性及渐近分析

论文总字数：28717字

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\begin{center}{\songti \sanhao{一类热传导方程初边值问题解的适定性及渐近分析}}\end{center}

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\begin{center}{\songti \xiaosihao \bf 摘\ \ \ \ 要}

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\songti{

本文研究一类热传导方程初边值问题，这类问题源于扩散光学层析成像（DOT）。DOT~是一项新兴的利用近红外光照射获得三维图像的医学成像技术，以其无损性、信息特异性、低成本性等特点而广受人们关注。这种使用近红外光为激发源，通过测量组织表面的光强分布信息，并采用一定的重构算法重建生物组织光学参数的二维和三维分布的技术，已在许多领域广泛应用于临床诊断。

假设扩散介质中含有小尺寸异常物质，此时~DOT~的扩散模型为一类带间断系数的热方程混合初边值问题。本文将运用变分方法分析定解问题的适定性，并根据偏微分

方程的定性理论研究当扩散介质中异常物体的直径趋于零时解的渐近行为，最后利用有限元方法求定解问题的数值解，从数值上验证理论结果。}

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\vskip 1cm \noindent{\xiaosihao {\bf 关键词:} \ 扩散光学层析成像，热传导方程，小异常物质，适定性，渐近性 }

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%\begin{center}{\The fitness and asymptotic analysis of the solutions to the initial boundary value problems of a class of heat conduction equations}\end{center}

\begin{center}{\centering\sanhao\bfseries The well-posedness and asymptotic analysis of the solutions to the initial boundary value problems of a class of heat conduction equations}\end{center}

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\songti

We study a class of initial boundary value problems of heat conduction equation, which are commonly used in diffusion optical tomography (DOT). DOT~ is a new medical imaging technology that uses the near-infrared irradiation to get three-dimensional images

and is well-know for its nondestructive, information specific, low cost and other characteristics. Using the near-infrared light as the excitation source, measuring the information of the light intensity distribution on the tissue surface, and using a certain reconstruction algorithm to reconstruct the two-dimensional and three-dimensional distribution of the biological tissue optics parameters, this technology has been widely used in clinical applications in many fields.

We assume that the diffusion medium contains small inclusions, then the diffusion model of ~DOT~ becomes a kind of mixed initial boundary value problem with discontinuous coefficient. So we use the variational method to verify the well-posedness of the problem and prove the asymptotic behavior of solution when the diameter of inclusions approaches to zero according to the study of partial differential equation qualitative theory. Finally, we use the finite element method to solve the numerical solution of the problem and present some numerical results to verify the theoretical results.

\vskip 1cm \noindent{\xiaosihao {\bf Keywords:} DOT, \ heat conduction equation, \ small inclusions, \ well-posedness, \ asymptotic behavior}

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