| 时间分数阶扩散方程的微分阶数识别 |
| Alternative Title | Determine of order for the time-fractional diffusion equation
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| 李文 |
| Thesis Advisor | 魏婷
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| 2016-05-21
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| Degree Grantor | 兰州大学
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| Place of Conferral | 兰州
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| Degree Name | 硕士
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| Keyword | 多项时间分数阶
Mittag-Leffler函数
阶数识别
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| Abstract | 本文中，我们考虑多项时间分数阶扩散方程的时间分数阶阶数识别问题，即由内部一点上的观测数据去反演Caputo导数阶数。关于正问题的数值解法，我们利用有限差分方法，给出求解正问题的隐式差分格式，通过对系数矩阵元素的分析，证明差分格式的无条件稳定性及收敛性。关于阶数识别方面，我们利用预备知识中给出了多项 Mittag-Leffler 的重要性质去分析正问题的分离变量解在时间方向的渐近性，借助于上述渐近性分析，我们给出了阶数反演的一个显示表达式。在数值实验方面，我们借助前面给出的差分格式来得到观测数据，进而去验证上述显示表达式在阶数识别上的有效性。 |
| Other Abstract | In this paper, we consider an inverse problem to determine the unknown order of Caputo fractional derivative of a multi-term time fractional diffusion equation with a Robin boundary condition. The implicit numerical method is employed to solve the direct problem, the stability and convergence of the proposed method are discussed. For the inverse problem, we first give some important properties of the multi-term Mittag-Leffler function, by using these properties we obtain the formulae of reconstructing the orders of time fractional derivative in the fractional diffusion equation by time history at one fixed spatial point. At last, we show numerical tests for our reconstruction formula. |
| URL | 查看原文
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| Language | 中文
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| Document Type | 学位论文
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| Identifier | https://ir.lzu.edu.cn/handle/262010/224716
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| Collection | 数学与统计学院
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Recommended Citation
GB/T 7714 |
李文. 时间分数阶扩散方程的微分阶数识别[D]. 兰州. 兰州大学,2016.
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