兰州大学机构库 >数学与统计学院
分数阶扩散方程正问题和反问题的理论及数值方法研究
Alternative TitleTheoretical and numerical method studies on the direct and inverse problems of fractional diffusion equations
姜素珍
Subtype博士
Thesis Advisor伍渝江
2023-05-20
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name理学博士
Degree Discipline计算数学
Keyword时间分数阶扩散方程 time fractional diffusion equation $L1$ 方法 $L1$ scheme $L2-1\sigma$方法 $L2-1\sigma$ scheme 分级网格 graded meshes 稳定性和收敛性 stability and convergence 唯一性 uniqueness 非定常迭代Tikhonov正则化方法 non-stationary iterative Tikhonov regularization method Levenberg–Marquardt方法. Levenberg-Marquardt method.
Abstract

本文中我们主要考虑了时间分数阶扩散方程正问题和反问题的理论及数值方法研究: 在分级网格上用两种 $L1$方法数值计算时间分数阶 Feynman-Kac方程; 在分级网格上用两种 $L2-1\sigma$方法数值计算时间分数阶 Feynman-Kac方程; 多项时间分数阶扩散方程中的依赖空间变量的源项函数辨识问题;利用非线性条件修复多项时间分数阶扩散方程中的依赖时间变量的势函数.

本文的第一部分我们主要考虑了两类$L1$方法数值计算时间分数阶Feynman-Kac方程, 其基于分段线性插值近似一阶时间导数. 在分级网格下, 通过隐式 $L1$方法收敛阶下降至$\mathcal{O}\left(\tau^{\min\{1,~r\alpha\}}\right).$ 这促使我们设计隐式- 显式 $L1$ 方法, 这样设计的格式在分级网格下可以达到最优收敛阶 $\mathcal{O}\left(\tau^{\min\{2-\alpha,~r\alpha\}}\right).$ 我们给出了两种$L1$方法稳定性和收敛性的证明. 最后通过一维和二维算例验证了所提出方法的可行性和有效性.

本文的第二部分我们主要考虑了两类$L2-1\sigma$方法数值计算时间分数阶Feynman-Kac 方程, 其基于分段二次插值近似一阶时间导数. 我们设计的隐式$L2-1\sigma$方法和修正的隐式$L2-1\sigma$方法, 它们的收敛阶是$\mathcal{O}\big(\tau^{\min\{1, r\alpha\}}\big).$ 这就促使我们设计另一个修正的隐式$L2-1\sigma$ 方法, 这样设计的格式在分级网格上可以达到最优收敛阶$\mathcal{O}\big(\tau^{\min\{2, r\alpha\}}\big).$ 我们给出了修正的隐式$L2-1\sigma$ 方法的稳定性分析. 最后通过二维数值算例验证了所提方法的可行性和有效性.

本文的第三部分主要考虑了从带噪声的终端数据重构多项时间分数阶扩散方程中的依赖空间变量的源项. 首先, 利用Caputo导数算子和椭圆算子的性质, 我们证明了正问题的解存在性和唯一性. 其次, 利用Caputo导数算子的性质和傅里叶变换, 我们证明了依赖空间变量源项辨识问题的唯一性. 最后我们运用非定常迭代Tikhonov 正则化方法结合有限维近似数值求解, 通过四个不同数值例子验证了所提出的数值方法是可行的和有效的.

本文的第四部分我们主要致力于了用空间区域上的积分测量数据重构多项时间分数阶扩散方程中的依赖时间变量的势函数. 首先, 我们利用不动点定理, 证明了正问题的解存在性, 唯一性和正则性.  其次, 利用Caputo导数的性质我们证明了反问题的唯一性. 最后我们用Levenberg-Marquardt方法进行数值计算势函数, 四个数值例子表明我们提出的数值方法是可行的和有效的.

Other Abstract

In this thesis, we mainly consider  theoretical and numerical methods of direct and inverse problems for time fractional diffusion equations: two $L1$ schemes on graded meshes for time fractional Feynman-Kac equation; two $L2-1\sigma$ schemes on graded meshes for time fractional Feynman-Kac equation; The problem of space dependent source term identification for multi-time fractional diffusion equation;
recovering a time-dependent potential function in a multi-term time fractional diffusion equation by using a nonlinear condition.

The first part of this thesis, we mainly consider the time fractional Feynman-Kac equation is numerically calculated by two kinds of $L1$ scheme, which approximates the first time derivative based on piecewise linear interpolation.
Its convergence order shall drop down to $\mathcal{O}\left(\tau^{\min\{1,~r\alpha\}}\right)$ by the implicit $L1$ scheme on graded meshes. This motivates us to
design the implicit-explicit $L1$ scheme, which reaches the optimal convergence order $\mathcal{O}\left(\tau^{\min\{2-\alpha,~r\alpha\}}\right)$ on graded meshes. We prove the stability and convergence of two $L1$ schemes. Finally, the feasibility and effectiveness of the proposed method are verified by one and two dimensional examples.

In the second part of this thesis, we mainly consider the
time fractional Feynman-Kac equation is numerically calculated by two kinds of $L2-1\sigma$ scheme, which approximates the first time derivative based on piecewise quadratic interpolation. We design the implicit $L2-1\sigma$ scheme and modified implicit $L2-1\sigma$ schemes, their order of convergence is  $\mathcal{O}\left(\tau^{\min\{1,~r\alpha\}}\right).$ This motivates us to design another modified implicit $L2-1\sigma$ scheme, which reach the optimal convergence order $\mathcal{O}\left(\tau^{\min\{2,~r\alpha\}}\right)$ on graded meshes. We give the stability analysis of the modified implicit $L2-1\sigma$ method.
Finally, the feasibility and effectiveness of the proposed method are verified by two dimensional example.

In the third part of this thesis, we mainly consider recovering the space-dependent source for a multi-term time fractional diffusion equation from noisy final data. Firstly, we proved
that the direct problem has a unique solution by the properties of the Caputo derivative operator and elliptic operator. Secondly, using the properties of the Caputo derivative operator and Fourier transform, we proved the uniqueness of the  space-dependent source term identification problem.
 Finally, we apply a non-stationary iterative Tikhonov regularization method combined with a finite dimensional approximation numerical solution. Four different examples are presented to show the feasibility and efficiency of the proposed method.

In the fourth part of this thesis,  we focus on the reconstruction of the  time-dependent potential function in a multi-term time fractional diffusion equation from an additional measurement in the form of an integral over the space domain. Firstly, we prove the existence, uniqueness and regularity  of the solution for the direct problem by using the fixed point theorem. Secondly, we prove the uniqueness of the inverse problem by using the property of Caputo derivative.
Finally, we use Levenberg-Marquardt method to simulate the potential function. Four examples show that the proposed numerical method is feasible and effective.

MOST Discipline Catalogue理学 - 数学 - 计算数学
URL查看原文
Language中文
Other Code262010_120190905840
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/534742
Collection数学与统计学院
Affiliation
兰州大学数学与统计学院
Recommended Citation
GB/T 7714
姜素珍. 分数阶扩散方程正问题和反问题的理论及数值方法研究[D]. 兰州. 兰州大学,2023.
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