兰州大学机构库 >数学与统计学院
时间分数阶扩散方程源项和系数辨识问题研究
Alternative TitleStudies on determination of source and coefficient for the time-fractional diffusion equations
孙亮亮
Subtype博士
Thesis Advisor魏婷
2017-10-01
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword时间分数阶扩散方程 反源问题 零阶项系数 分数阶阶数 对流项系数 非线性反问题 多项Mittag-Leffler 函数 广义的Gronwall 不等式 Tikhonov 正则化 共轭梯度法 最佳慑动算法 存在唯一性 稳定性 收敛性
Abstract

近年来, 受实际问题的驱动, 时间分数阶扩散方程 (TFDEs) 引起了广泛的关注. 关于 TFDEs 正问题的研究已经取得了很大的进展, 然而对TFDEs 反问题的研究, 包括理论分析和数值计算都还处于初级阶段. 鉴于以上现状, 本文主要考虑 TFDEs 中的以下几类反问题.

第一部分考虑了一类 TFDEs 中依赖空间变元的源项识别问题. 我们主要考察高维空间一般有界性区域上由边界数据反演源项的唯一性.
首先我们利用分离变量法证明了正问题弱解的存在唯一性, 其次利用 Laplace 变换, 证明了反源问题的唯一性. 第二部分考虑了一类 TFDEs 中仅依赖空间变元的零阶项系数辨识问题. 我们主要考察在一维情形下由边界数据反演零阶项系数的唯一性及数值算法, 这是一个非线性反问题. 首先我们定义了正问题两种类型的弱解, 并证明其存在唯一性. 其次通过 Laplace 变换和 Gel'fand-Levitan 理论证明了同时反演零阶项系数和分数阶阶数的唯一性. 然后利用 Tikhonov 正则化将反演系数问题转化成一个变分问题, 并得到了该变分问题解的存在性, 稳定性和收敛性. 最后利用共轭梯度法求解了该反问题, 其数值结果表明我们的方法是有效的. 第三部分考虑了一类多项时间分数阶扩散方程 (MTFDEs) 中依赖空间变元的源项识别问题. 这是第一部分的一个推广. 我们主要考察在高维空间由边界数据反演源项的唯一性. 首先我们利用变分的方法导出正问题的一个弱形式, 并由 Lax-Milgram 定理证明了该弱形式解的存在唯一性, 其次利用 Laplace 变换和多项 Mittag-Leffler 函数的相关性质, 获得了反源问题的唯一性. 第四部分考虑了一类 MTFDEs 中仅依赖空间变元的零阶项系数和分数阶阶数辨识问题, 这是第二部分的一个推广. 我们主要考察了一维情形下由边界数据同时反演零阶项系数和分数阶阶数的唯一性和数值重构. 首先我们给出了正问题的一个弱形式, 并证明其弱解的存在唯一性. 其次通过 Laplace 变换和 Gel'fand-Levitan 理论和解析延拓技巧证明了反问题的唯一性. 然后利用 Tikhonov 正则化将该反问题转化成一个变分问题, 并用交替迭代方法求解了该变分问题. 借助于灵敏度问题和共轭问题, 我们导出了求解零阶项系数的共轭梯度算法, 并用最佳慑动算法求解了分数阶阶数, 数值结果表明我们的算法是有效的. 第五部分考虑了一类时间分数阶对流-扩散方程 (TFCDE) 中依赖时间变元的对流项系数识别问题, 这同样也是一个非线性的反问题. 我们主要考察了一维情形中由内部点测量数据反演对流项系数的某种稳定性和数值重构. 首先我们利用不动点定理证明了对应正问题解的存在唯一性和正则性. 然后基于正问题解的正则性和广义的 Gronwall 不等式, 我们得到了反演对流项系数问题的稳定性. 最后利用带有 Sigmoid 型函数的正则化参数的最佳慑动算法求解了该反系数问题, 数值算例表明我们的算法是有效的。

Other Abstract

In recent years, the time fractional diffusion equations (TFDEs) have attracted wide attentions motivated by practical problems. The studies on the direct problems for the time-fractional diffusion equations have made great progress. However, the studies on the inverse problems of TFDEs, including theoretical analysis and numerical computation is still in its infancy. In view of the above situation, we discuss the following classes of inverse problems for the time-fractional diffusion equation in the present paper.

In Part 1, we discuss a space-dependent source identification problem in a time-fractional diffusion equation, and mainly study the uniqueness of inverse source problem in a bounded domain of multidimensional case. The existence and uniqueness of the weak solution for the direct problem is proved by the separation variable method. Then we prove the uniqueness for the inverse source problem by the Laplace transformation. In Part 2, we study an identification of the space-dependent zeroth-order coefficient in a time-fractional diffusion equation, and mainly investigate the uniqueness of determining the zeroth-order coefficient and its numerical reconstruction from two boundary measurement data in one-dimensional case, which is a nonlinear inverse problem. The existence and uniqueness of two kinds of weak solutions for the direct problem are proved. Then we provide the uniqueness for recovering the zeroth-order coefficient and fractional order simultaneously by the Laplace transformation and the Gel'fand-Levitan theory. The identification of the zeroth-order coefficient is formulated into a variational problem by the Tikhonov regularization. The existence, stability and convergence of the solution for the variational problem are provided. Finally, the conjugate gradient method is used to solve it. Numerical examples are provided to show the effectiveness of our method. In Part 3, we devote our efforts to an inverse space-dependent source problem in a multi-term time-fractional diffusion equation from boundary measured data, which is an extension of Part 1, and obtain the uniqueness of inverse source problem in a bounded domain of multidimensional case. First, we give a weak formulation by the variational method for the direct problem, and obtain the existence and uniqueness of a weak solution for the direct problem by using the well-known Lax-Milgram theorem. Then by employing the Laplace transformation and related properties of multinomial Mittag-Leffler function, the uniqueness for the inverse source problem is derived. In Part 4, we investigate a nonlinear inverse problem for identifying the space-dependent zeroth-order coefficient and fractional orders in a multi-term time-fractional diffusion equation, which is an extension of Part 2, and we mainly discuss the uniqueness for recovering the zeroth-order coefficient and fractional orders simultaneously and its numerical reconstruction from boundary measurement data in one-dimensional case. First, we give a weak formulation for the direct problem, and obtain the existence and uniqueness of a weak solution for the direct problem. Then we provide the uniqueness of the inverse problems by the Laplace transformation, the Gel'fand-Levitan theory and analytic continuation skills. Finally, the identification of the zeroth-order coefficient and the fractional orders is formulated into a variational problem by the Tikhonov regularization, and the variational problem is solved by an alterative iteration method. With the help of sensitivity problem and adjoint problem we use a conjugate gradient method to find the approximate zeroth-order coefficient and use the optimal perturbation algorithm to recover the fractional orders. Numerical examples are provided to show the effectiveness of the proposed method. In Part 5, we devote our effort to a nonlinear inverse problem for identifying a time-dependent convection coefficient in a time-fractional convection diffusion equation, and we mainly obtained the stability for recovering the convection coefficient and its numerical reconstruction from the measured data at an interior point for one-dimensional case. We prove the existence, uniqueness and regularity of solution for the direct problem by using the fixed point theorem. The stability of inverse convection coefficient problem is obtained based on the regularity of solution of direct problem and some generalized Gronwall's inequalities. We use the optimal perturbation algorithm with a regularization parameter chosen by a sigmoid-type function to solve the inverse convection coefficient problem. Two numerical examples are provided to show the effectiveness of the proposed method.

URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224712
Collection数学与统计学院
Recommended Citation
GB/T 7714
孙亮亮. 时间分数阶扩散方程源项和系数辨识问题研究[D]. 兰州. 兰州大学,2017.
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