| Abstract | 在本文中, 我们主要考虑了分数阶偏微分方程中的几类反问题, 例如时间分数阶逆对流扩散问题(TFIADP), 时间分数阶对流扩散Cauchy 问题(TFADCP), 时间分数阶扩散Cauchy 问题(TFDCP), 时间分数阶逆扩散问题(TFIDP) 以及空间分数阶逆时扩散问题(SFBDP). 随后, 还考虑了多层区域上抛物型系统在线性和非线性接触条件下的边界识别问题(BIP), 以及一般抛物方程中同时反演源项和初值的问题. |
| Other Abstract | In this thesis, we consider several inverse problems in fractional PDEs, such as time fractional inverse advection-dispersion problem (TFIADP), time fractional advection-dispersion Cauchy problem (TFADCP), time fractional diffusion Cauchy problem (TFDCP), time fractional inverse diffusion problem (TFIDP) and space fractional backward diffusion problem (SFBDP). Moreover, we consider boundary
identification problems (BIP) in the parabolic system with a multi-layer domain, which refer to linear and nonlinear interface conditions, and recovering the source and initial value simultaneously in a parabolic equation.
Fractional PDEs have been used recently to describe a range of problems in physical, chemical, biology, finance, signal processing, systems identification, control theory and so on. As for direct problems in fractional PDEs, there are a lot of researches both in fundamental theory and numerical computation. However, the result for the corresponding inverse problems is still very sparse. In this thesis, we propose a new convolution-type regularization method to solve time fractional inverse advection-dispersion problem (TFIADP), time fractional diffusion Cauchy problem (TFDCP), time fractional inverse diffusion problem (TFIDP), space fractional backward diffusion problem (SFBDP), and convergence estimates of the
regularization method are presented. Furthermore, we apply spectral regularization method to solve the time fractional inverse advection-dispersion problem (TFIADP), time fractional advection-dispersion Cauchy problem (TFADCP), space fractional backward diffusion problem (SFBDP) and obtain the corresponding convergence estimates. Finally, we make numerical tests for above two regularization methods to show the effectiveness.
The boundary identification problem (BIP) is very important in the research area of inverse problems in PDEs, and has wide application background. Because of ill-posedness and nonlinearity, the boundary identification problems are very challenging. Here, we obtain the stability estimates and uniqueness for the boundary identification problems in the parabolic system with a multi-layer domain.
Under some given measurement data, it is a hot topic to recover simultaneously two goals, even more goals in inverse problems. Because it need to recover more goals, it is harder than conventional inverse problems. We investigate recovering the source and initial value simultaneously in a parabolic equation. Stability esti... |
