分数阶扩散方程反问题的再生核方法

Numerical method and its application are investigated for fractional diffusion equation, which are new mathematical aspects motivated by industrial collaboration. Recently, anomalous diffusion phenomena have been observed in a broad variety of engineering and physics fields, such as electron transportation, heat conduction, magnetic plasma, etc. In particular, anomalous diffusion is seriously concerned with environmental problems, such as evaluating the density of underground contaminants. Such anomalous diffusion phenomena can be effectively modeled by fractional diffusion equations, which have attracted intense attention in recent years. We will construct the reproducing kernel space according to the fractional diffusion equation. In order to improve the precision and decrease the runtime, we will obtain more simple reproducing kernels as possible as we can. In the reproducing kernel space, we will apply the regularization method to inverse the fractional diffusion equation. This topic is the frontier's investigation in the world, which attracts more attention in engineering, physics and mathematics fields. Few reports appeared in China. The project conducted has important significance for developing and understanding the mathematical therories and algorithms of anomalous diffusion phenomena, as well as provides an important support for environmental problems such as evaluation of underground contaminants.

本项目主要研究分数阶扩散方程的数值解法及其应用，这是产学合作推动的新研究领域。近年来,异常扩散现象在工程学和物理学等许多领域出现,例如电子通讯、热传导、磁等离子体等。特别地，异常扩散与环境问题密切相关, 例如地下污染的评估。分数阶扩散方程可以有效地模拟这种异常扩散现象, 因此引起了广泛的关注。本项目将构造适宜分数阶扩散方程的再生核空间，将尽可能地获得较简单的再生核，从而提高运算精度，减少运算时间；将在此再生核空间中，用正则化方法反演分数阶扩散方程。此研究属国际前沿性课题，受到国际工程学界、物理学界和数学界的广泛关注，国内在该方向的研究较少。本项目的开展，对发展和理解异常扩散问题的数学理论和数值算法具有重要意义，也将对环境问题例如地下污染的评估起到重要的支撑作用。

本项目主要研究分数阶扩散方程的数值解法及其应用，这是产学合作推动的新研究领域。近年来,异常扩散现象在各种科学和工程领域出现,例如电子通讯、热传导、系统控制、溶质运移、混沌等。特别地，异常扩散与环境问题密切相关, 例如地下污染的评估。分数阶扩散方程可以有效地模拟这种异常扩散现象, 因此引起了广泛的关注。本项目构造了适宜分数阶扩散方程的再生核空间，获得了较简单的再生核，提高了运算精度，减少了运算时间；并在此再生核空间中，用正则化方法反演了分数阶扩散方程。此外，结合分段多项式再生核和多项式再生核的优点，构造出优化的再生核法。与经典的再生核法相比较，该方法具有较高的精度，并且不需要将非齐次边界条件转化为齐次边界条件，使得再生核算法得到了进一步的改进。与域分解方法、有限元法和激光视觉测量法相结合，进一步地扩大了再生核法的应用领域。