【作者】 王宇；

【导师】 孙伟；

【作者基本信息】 哈尔滨理工大学 ， 数学， 2024， 硕士

【摘要】 热传导方程反问题作为传热学、物理、数学、计算机等多个学科交叉的领域,它的研究具有深远的意义。本文主要研究了矩形区域和柱对称区域上热传导方程以及带有记忆项热传导方程反问题的正则化方法。首先研究了二维矩形域上热传导方程的逆时问题,考虑只与空间位置有关源项和终值数据同时带有测量误差的情况,利用源项和终值数据的Fourier展开,构造了一种正则化方法,此方法是一种已有正则化方法的扩展和改进;研究了几种不同的先验假设下正则化参数的先验和后验选取策略,并分析了正则化解的收敛性;数值算例验证了该方法的有效性及可行性。其次探讨了柱对称区域上热传导方程的源项反演问题,构造一种基于Fourier展开的正则化方法,在两种不同的先验假设下,得到了正则化参数的先验选取规则和近似解的误差估计;随后,利用相同的思想求解了柱对称区域上热传导方程源项和初值同时反演的问题,提供了正则化参数的先验选取策略并进行收敛性分析;同时利用数值算例验证了正则化方法的有效性。最后考虑了一类带有记忆项热传导方程的逆时问题,采用滤波正则化方法求解此问题,阐述了滤波器满足的条件,同时给出了正则化参数先验和后验选取策略以及L2范数意义下的误差估计;此外,探讨了一类特殊的滤波器,不仅研究了此种情况下正则化解在L2范数意义下的收敛性,还讨论了其在Hq范数意义下的收敛性;通过数值算例证明了方法的可行性。更多还原

【Abstract】 The inverse problem of the heat conduction equation is significant as a cross field,including heat transfer,physics,mathematics,and computers.This paper focuses on regularization methods for the inverse problems of heat conduction equations on a rectangular and a column radially symmetric domain,as well as for the inverse problem of heat conduction equation with memory.Firstly,the backward heat conduction problem on a two-dimensional rectangular is investigated.Considering the case where both the space-dependent source and the final value data simultaneously have measurement errors,a regularization method is constructed by using the Fourier expansion of the source and the final value data.This method is an extension and improvement of a known regularization method.Priori and posteriori selection strategies of the regularization parameter are investigated under different priori assumptions,and the convergences of the regularization solution are analyzed.Numerical examples verify the effectiveness and feasibility of the method.Secondly,the source identification problem of heat conduction equation on a column radially symmetric domain is explored.A regularization method based on Fourier expansion is constructed.The prior selection rules of the regularization parameter and the error estimates of approximate solution are obtained under two different prior assumptions.Subsequently,the simultaneous inversion of the source and the initial value for the heat conduction equation on a column radially symmetric domain is solved by using the same idea.A prior selection strategy for the regularization parameter is provided,and the convergence is analyzed.Numerical examples are also presented to confirm the effectiveness of the regularization method.Finally,the backward problem for the heat conduction equation with memory is considered.A filter regularization method is used to solve this problem.The conditions satisfied by the filter are stated.Simultaneously,both the prior and posterior selection strategies for the regularization parameter are presented,as well as the error estimations in the sense of the L2 norm.Moreover,a class of special filters is explored.Not only are the convergences of the regularization solution investigated in the sense of the L2 norm,but the convergences are discussed in the sense of the Hq norm.Numerical examples show that the method is feasible.更多还原