Recovering Robin Coefficient For The Time-fractional Diffusion Equation

Diffusion is an important type of physical phenomenon,which exists widely in the fields of natural sciences and industrial applications.The physical properties,initial state and boundary information of the diffusion medium determine the distribution law and dynamic evolution process of related physical quantities.In many practical problems,the information on the boundary is unknown.Then it is necessary to recover the unknown information of the boundary through extra measurement data to determine the entire diffusion process.This paper studies the diffusion process described by a class of time fractional diffusion equation with Robin boundary conditions,and reconstructs the Robin coefficients of the medium from boundary measurement data.This is a typical nonlinear ill-posed problem.Although the Robin coefficient inverse problems of integer-order diffusion equations have many profound results,the fractional-order situation needs further study.In this paper,we reconstruct the space-dependent Robin coefficient of the time fractional diffusion equation in the two-dimensional space by the weighted boundary measurement data.This paper is divided into five chapters:In Chapter 1,we introduce the background and related works on inverse problems for the time fractional diffusion equation.Then our novel work of this paper is described.In Chapter 2,we study the forward problem for the time fractional diffusion equation.First,we give some necessary preliminary knowledge on the boundary integral equation method.Then,using the potential theory,the problem is converted into a boundary integral equation.Finally,considering the weak singularity of the single-layer layer and its normal derivative,a discretization scheme is proposed.In Chapter 3,we study the Robin coefficient inverse problem.A nonlinear integral system is derived by using the boundary measurement data and potential theory,from which the uniqueness and stability of the solution are proved under certain conditions.Then,the regularization method is used to construct cost function.We prove the well-posedness of the functional and the approximation property of the minimizer of the functional.Finally,an iterative algorithm is given to solve the minimizer of the functional,and a numerical discretization scheme is designed.In Chapter 4,several numerical examples are presented to verify the effectiveness of the numerical schemes proposed in Chapter 2 and Chapter 3.In Chapter 5,we briefly summarize the research work of the whole paper and prospects the related work in the future.