Studies For Two Kinds Of Inverse Problems For Fractional Diffusion Equations

During the past three decades,fractional diffusion equations arise in various scientific and engineering problems related to anomalous diffusion,which might be inconsistent with the classical Brownian motion model.The inverse problems for fractional diffusion equation have been considered by many researchers in many theoretical papers.This thesis considers two kinds of inverse problems for fractional diffusion equations,i.e.,identifying a diffusion coefficient in a time-fractional diffusion equation,an inverse time-dependent source problem for a multi-term time-fractional diffusion equation and a time-space fractional diffusion equation.In part 1,we propose a conjugate gradient algorithm for identifying a spacedependent diffusion coefficient in a time-fractional diffusion equation from the boundary Cauchy data in one-dimensional case.The existence and uniqueness of the solution for a weak form of the direct problem are obtained.The identification of diffusion coefficient is formulated into a variational problem by the Tikhonov-type regularization.The existence,stability and convergence of a minimizer for the variational problem approaching to the exact diffusion coefficient are provided.We use a conjugate gradient method to solve the variational problem based on the deductions of a sensitive problem and an adjoint problem.We test three numerical examples and show the effectiveness of the proposed method.Part 2 aims to identify a time-dependent source term in a multi-term timefractional diffusion equation from the boundary Cauchy data.The regularity of the weak solution for the direct problem with homogeneous Neumann boundary condition is proved.We provide the uniqueness and a stability estimate for the inverse timedependent source problem.On the other hand,the inverse time-dependent source term is formulated into a variational problem by the Tikhonov regularization,with the help of sensitivity problem and adjoint problem we use a conjugate gradient method to find the approximate time-dependent source term.Numerical experiments for five examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.Part 3 is devoted to identify a time-dependent source term in a time-space fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point.The existence and uniqueness of a weak solution for the corresponding direct problem with homogeneous Dirichlet boundary condition are proved.We provide the uniqueness and a stability estimate for the inverse time-dependent source problem.Based on the separation of variables,we transform the inverse source problem into a first kind Volterra integral equation with the source term as a unknown function and then show the ill-posedness of the problem.Further,we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the first kind.The generalized cross validation rule for the choice of regularization parameter is applied to obtain a stable numerical approximation to the time-dependent source term.Numerical experiments for six examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.