基于分裂Bregman迭代方法的时间分数阶扩散波方程 空间源项辨识问题 Alternative Title )The split Bregman iteration method for determining the space-dependent source function in a time-fractional diﬀusion-wave equation 刘异 Subtype 硕士 Thesis Advisor 魏婷 2020-05-17 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 理学硕士 Degree Discipline 应用数学 Keyword 反空间源项 时间分数阶扩散波方程 分裂Bregman迭代方法 Abstract 本文主要研究时间分数阶扩散波方程非光滑空间源项的辨识问题。首先由正问题解 的级数表达式，将反问题转化为求解第一类Fredholm积分方程，讨论反问题的解的存在 唯一性及不适定性。在本文中，考虑一种L2(Ω)与TV 混合罚项的Banach空间的正则化方 法，将反问题转化为正则化变分问题，给出正则化解的存在唯一性，稳定性，收敛性，收 敛阶分析。然后通过分裂Bregman迭代方法来快速获得近似解。最后，我们分别给出几个 一维和二维的数值算例来展示算法的有效性 Other Abstract This paper mainly considers the inverse non-smooth space-dependent source problem for a time-fractional diﬀusion-wave equation. Firstly, based on the series expression of the solution for the direct problem, we transform the inverse problem into solving a Fredholm integral equation of the ﬁrst kind, then we discuss the uniqueness and ill-posedness of the inverse problem. In this paper, we use a Banach regularization method of mixed penalty terms with L2(Ω) and TV for converting the inverse problem into a regularized variational problem and give the existence and uniqueness of regularization solution, convergence, and analysis of convergence order. Then we obtain an approximate solution quickly by using the split Bregman iteration method. In the end, numerical results for several examples in one-dimensional and two-dimensional cases show that the algorithm is eﬃcient. Pages 40 URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/464185 Collection 数学与统计学院 Affiliation 数学与统计学院 First Author Affilication School of Mathematics and Statistics Recommended CitationGB/T 7714 刘异. 基于分裂Bregman迭代方法的时间分数阶扩散波方程 空间源项辨识问题[D]. 兰州. 兰州大学,2020.
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