求解一类拟逆正则化问题的数值计算方法 Alternative Title numerical method for solving a quasi-reversibility regularized problem 李欣杭 Subtype 学士 Thesis Advisor 魏婷 2023-05-18 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 理学学士 Degree Discipline 数学与应用数学 Keyword 时间分数阶扩散波方程 time-fractional diffusion-wave equation 拟逆正则化方法 quasi-reversibility method 有限差分方法 finite difference method 有限元方法 finite element method Abstract 在本文中, 我们主要研究有界区域上的时间分数阶扩散波方程空间源项反演问题经一种拟逆正则化方法转化后得到的正则化问题的数值求解方法. 时间分数阶扩散波方程空间源项反演问题即根据带误差的终端数据来反演空间源项. 本文首先基于正问题的解析解将反问题转为第一类 Ferdholm 积分方程, 进而给出了反问题解的唯一性、不适定性和条件稳定性. 之后本文引入了一种拟逆正则化方法将该问题转化为了一种正则化问题, 讨论了正则化问题的适定性. 接着本文提出了一种有限差分与有限元结合的方法对该正则化问题进行了数值求解, 并在最后初步验证了结果的正确性. Other Abstract In this paper, we mainly focus on the numerical method for solving the problem of identifying a space-dependent source term for a time-fractional diffusion-wave equation in a bounded domain, which means using the final time noisy data to recover the space-dependent source. First, the inverse problem can be formualted into a first kind of Ferdholm integral equation based on the analytical solution of the direct problem. Then, we discuss the uniqueness, ill-posedness and conditional stability of the inverse problem. After that, a quasi-reversibility regularization method is introduced to transform the inverse problem into a regularized problem, and the well-posedness of the regularized problem is discussed. Finally, we propose a method, a combination of finite difference method for time variable and finite element method for space variable, to solve the regularized problem numerically, and verified the correctness of the results. MOST Discipline Catalogue 理学 - 数学 - 计算数学 URL 查看原文 Language 中文 Other Code 262010_320190927401 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/537047 Collection 数学与统计学院 Affiliation 兰州大学数学与统计学院 Recommended CitationGB/T 7714 李欣杭. 求解一类拟逆正则化问题的数值计算方法[D]. 兰州. 兰州大学,2023.
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