Regularization Metnoas For Some Inverse Problems Of Fractional PDEs

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, con-trol 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 frac-tional backward diffusion problem (SFBDP), and convergence estimates of the regularization method are presented. Furthermore, we apply spectral regulariza-tion method to solve the time fractional inverse advection-dispersion problem (T-FIADP), time fractional advection-dispersion Cauchy problem (TFADCP), space fractional backward diffusion problem (SFBDP) and obtain the corresponding con-vergence 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 bound- ary identification problems in the parabolic system with a multi-layer domain.Under some given measurement data, it is a hot topic to recover simultane-ously two goals, even more goals in inverse problems. Because it need to recover more goals, it is harder than conventional inverse problems. We investigate recov-ering the source and initial value simultaneously in a parabolic equation. Stability estimate is given, then we apply variational Method to deal with it, and give ex-istence, uniqueness, stability of minimizer of minimizing functional. Moreover, convergence estimate is also given.