兰州大学机构库 >数学与统计学院
几类分数阶扩散方程反问题研究
Alternative TitleStudies on several kinds of inverse problems for fractional diffusion equations
张正强
Subtype博士
Thesis Advisor魏婷
2013-06-02
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword分数阶扩散方程 反问题 不适定问题 时间分数阶扩散方程Cauchy问题 时间分数阶扩散方程源项识别问题 Robin系数 空间分数阶反向扩散问题 Tikhonov正则化 收敛性估计
Abstract在实际问题的推动下,近些年分数阶扩散方程引起了广泛的关注,关于正问题的研究有了很大的进展.然而有时候因为部分边界上的数据不能直接得到或因为初值、源项、扩散系数未知,我们需要其他一些测量数据来求解这些未知量,这就产生了分数阶扩散方程反问题.本文考虑了以下几类分数阶扩散方程反问题. 第一部分考虑了时间分数阶扩散方程非特征~Cauchy~问题.利用分离变量法和~Duhamel~齐次化原理,我们将~Cauchy~问题转化成第一类的~Volterra~积分方程,证明了问题的不适定性,并利用边界元方法结合一阶~Tikhonov~正则化求解了此问题,数值例子表明我们给出的方法是有效的、稳定的. 第二部分别考虑了时间分数阶扩散方程中仅依赖时间变量或者空间变量的源项识别问题.对于依赖时间变量的源项识别问题,同样利用分离变量法和~Duhamel~齐次化原理将其转化成第一类的~Volterra~积分方程,证明了问题的不适定性,并利用边界元方法结合一阶~Tikhonov~正则化进行了数值模拟.对于只依赖空间变量的源项识别问题,我们将其转化成第一类的~Fredholm~积分方程,利用截断方法分别在先验和后验正则化参数选取规则下给出了收敛性分析. 第三部分考虑了时间分数阶扩散方程~Robin~系数识别问题.这是一个非线性问题,我们利用分离变量法将原问题转化成一个非线性积分方程组,通过边界元离散,最终将~Robin~系数识别问题转化成有限维空间的优化问题,并用共轭梯度法求解了此问题,数值结果表明我们给出的方法是有效的. 第四部分考虑了空间分数阶扩散方程反向问题.通过~Fourier~变换,我们在频域中给出一个最优的正则化方法,并分别在先验和后验正则化参数选取规则下给出了收敛性分析.数值例子验证了文中给出方法的有效性.
Other AbstractMotivated by practical problems, fractional diffusion equations haveattracted wide attentions in recent years and the direct problemshave been studied extensively. However, in some practicalsituations, part of boundary data, or initial data, or source term,or diffusion coefficients may not be given and we want to find themby additional measured data which will yield to some fractional diffusion inverse problems. This thesis discusses the following inverse problems for fractional diffusion equations. Part 1 discusses the Cauchy problem for the time fractional diffusion equation. Based on the separation of variables and Duhamel's principle, we transform the Cauchy problem into a first kind Volterra integral equation with the Neumann data as unknown function and then show the ill-posedness of problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the first kind integral equation. Numerical examples are provided to show the effectiveness and robustness of the proposed method. Part 2 studies the inverse source problem for the time fractionaldiffusion equation. For the source term depending on the time variable, based on the separation of variables and Duhamel's principle, we also transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem.Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the firstnkind. For source term depending on the space variable, we transform the inverse source problem into a first kind Fredholm integralnequation. A truncation method is presented to deal with the ill-posedness of the problem and error estimates are obtained with an \emph{a priori} choice rule and an \emph{a posteriori} choice rule to find the regularization parameter. Part 3, we consider a nonlinear inverse problem, i.e.,identifying a Robin coefficient for time fractional diffusion equation from part of the boundary data. Based on the separation of variables, we transform the problem into a nonlinear integral equations. By using the boundary element method, we finally obtain an optimization problem in finite dimensional space and the conjugate gradient method is used to solve it. Numerical examples are provided to show the effectiveness of our method. Part 4, a space-fractional backward diffusion problem (...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225047
Collection数学与统计学院
Recommended Citation
GB/T 7714
张正强. 几类分数阶扩散方程反问题研究[D]. 兰州. 兰州大学,2013.
Files in This Item:
There are no files associated with this item.
Related Services
Recommend this item
Bookmark
Usage statistics
Export to Endnote
Altmetrics Score
Google Scholar
Similar articles in Google Scholar
[张正强]'s Articles
Baidu academic
Similar articles in Baidu academic
[张正强]'s Articles
Bing Scholar
Similar articles in Bing Scholar
[张正强]'s Articles
Terms of Use
No data!
Social Bookmark/Share
No comment.
Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.