【作者】 熊向团；

【导师】 傅初黎；

【作者基本信息】 兰州大学 ， 应用数学， 2007， 博士

【摘要】 本论文研究了抛物型偏微分方程中的若干反问题，其中包括逆热传导问题、时间反向热传导问题、未知源确定问题等，分析了这几类问题的不适定本质和不适定程度，给出了各种正则化方法，尤其对谱正则化方法、小波对偶最小二乘法、有限差分方法作了系统的研究，给出了它们的误差估计。根据正则化理论，许多误差估计都是阶数最优的。本文分为六个部分，第一章前言简要分析了国内外抛物型偏微分方程反问题的研究现状；第二章数值微分的正则化及其应用从正则化理论和算法的角度出发，考察了许多正则化方法，还给出了数值微分在抛物型偏微分方程反问题的一些应用；第三章谱正则化方法是在Fourier分析的基础上，在一般正则化理论的框架下，给出了这种方法在各种不适定问题中的应用，数值实验表明谱方法是有效的；第四章研究了小波对偶最小二乘方法和改进的小波方法；第五章主要研究了有限差分方法结合线方法在时间反向热传导问题中的应用；第六章是未知源识别问题，主要指出了两类未知源问题的不适定程度和不适定本质，同时报告了一些数值方法。更多还原

【Abstract】 Some inverse problems for parabolic partial differential equations are studied in this thesis. These inverse problems include inverse heat conduction problems,backward heat conduction problems in time , identification problems for unknown source. We analyze the the essence and degree of these ill-posed problems and provide many regularization methods, especially we focus on the spectral regularization methods,wavelet dual least squares method and finite difference method, prove the error estimates for these methods as well. According to the general theory of regularization, many error estimates are order optimal.This thesis is divided into six parts. The first chapter is preface, the current status of research in the inverse problems for parabolic partial differential equations is reported; the second chapter is "regularization methods for numerical differentiation and their applications ", in this chapter we investigate many regularization methods from a viewpoint of regularization theory and algorithm, some applications in the inverse problems for parabolic partial differential equations are given; the third chapter is "spectral regularization methods". Based on Fourier analysis, within the framework of regularization theory, we apply the spectral methods to some ill-posed problems. Many numerical experiments are done in order to show the validity of the methods; the fourth chapter is devoted to wavelet dual least squares method and a revised wavelet method; in the fifth chapter,we combine finite difference method with method of lines and apply it to the backward heat conduction problem in time; in the sixth chapter "identification problems for unknown source ", the essence and the degree of two problems related to source identification are pointed out, at the same time, some numerical methods are reported.更多还原