兰州大学机构库 >数学与统计学院
时间分数阶扩散方程的两类不适定问题研究
Alternative TitleTwo kinds of ill-posed problems for time-fractional diffusion equation
王俊刚
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 反向问题 反源问题 Tikhonov正则化 拟边界值方法 迭代方法 拟逆正则化 收敛性估计
Abstract本文考虑了时间分数阶扩散方程的两类不适定问题: 包括时间分数阶扩散方程反向问题和时间分数阶扩散方程反源问题. 扩散方程反向问题是经典的不适定问题. 本文针对n维空间一般有界区域上的分数阶扩散方程反初值问题, 首先从理论上分析了该问题的不适定性, 给出阶最优的误差估计. 然后在此最优性框架指导下, 分别运用~Tikhonov 正则化方法, 简化~Tikhonov 正则化方法, 拟边界值正则化方法, 改进的拟边界方法和迭代方法来进行求解. 对这些正则化方法, 分别给出在先验正则化参数选取规则和后验正则化参数选取规则下的收敛性估计. 对一维问题, 用有限差分法来进行数值模拟, 对二维问题, 通过对时间进行差分, 对空间使用有限元离散来进行数值模拟. 对一般有界区域上的分数阶扩散方程反源问题, 分别运用~Tikhonov 正则化方法, 简化~Tikhonov 正则化方法和拟逆正则化方法来进行求解, 并分别给出在先验正则化参数选取规则和后验正 则化参数选取规则下的收敛性分析. 对一维问题用有限差分法来进行数值模拟, 对二维问题, 结合差分和有限元方法来进行数值模拟. 其中, 改进的拟边界方法和迭代方法都是很新颖有趣的方法. 而关于拟边界值正则化方法和拟逆正则化方法的后验参数选取规则都是第一次提出. 这些方法和结论都可以借鉴到别的问题当中, 进行更深入和广泛的研究. 本文的数值结果与理论结果是完全相符的, 这些数值结果充分地表明所给正则化方法能够很好地求解这些不适定问题.
Other AbstractIn this thesis, we consider two kinds of ill-posed problems for the time-fractional diffusion equation, i.e., the backward problem and the inverse source problem. The backward problem is a classical ill-posed problem. For the backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain, we first analyse the ill-posedness of the problem, then the optimal error bound for the problem under a source condition is obtained. We apply the Tikhonov regularization method, a simplified Tikhonov regularization method, the quasi-boundary value regularization method, a modified quasi-boundary value regularization method and an iteration method to deal with this problem. The corresponding convergence rates for these methods are analyzed under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule, respectively. For the numerical experiment, we use the finite difference method for 1-d case and the finite element method combined with finite difference method for 2-d case. In the second part of this thesis, the inverse problem of identifying a space-dependent source for the time-fractional diffusion equation is investigated. We apply the Tikhonov regularization method, a simplified Tikhonov regularization method and the quasi-reversibility method to solve it. The corresponding convergence estimates for these methods are obtained under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule, respectively. Many numerical examples are given to show that the regularization methods are effective and stable. Among all these works, the modified quasi-boundary value regularization method and an iteration method are two new and interesting methods. The a posteriori regularization parameter choice rules for the quasi-boundary value regularization method and the quasi-reversibility method are proposed for first time. These methods and results can be applied to other problems and be extensively studied further. The numerical results are consistent with the theoretical results. These results show that our regularization methods for these ill-posed problems work effectively.
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/224717
Collection数学与统计学院
Recommended Citation
GB/T 7714
王俊刚. 时间分数阶扩散方程的两类不适定问题研究[D]. 兰州. 兰州大学,2013.
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