Stability And Regularization Of Two Kinds Inverse Problems Of Time-space Fractional Diffusion Equation

In recent three decades, fractional differential equations have attracted wide atten-tions of scholars in mathematics and physics for its important application background. In certain situations, it is necessary to determine some immeasurable data, such as initial data, or source term, or diffusion coefficients or part of boundary data,i.e.,system iden-tification problem. This thesis mainly studies the following two kinds of system identi-fication problems for fractional diffusion equations, which includes the determination of coefficient for time-fractional diffusion equation and the identification of a source term for space-fractional diffusion equation.In the first two chapters, research background and main work of this paper, ill-posedness of inverse problems, regularization methods, two definitions of fractional deriva-tive and their Fourier transforms are introduced, respectively.Chapter 3 studies the determination of the conductivity coefficient in the time frac-tional diffusion equation (?)tα(x, t)-▽·γ(x)▽u(x, t)= 0, x ∈Ω(?) Rn,0< t< T, where 0< α≤ 1. We define a bilinear form function Qγ associated to the boundary condition and the observation of the lateral Dirichlet-to-Neumann map, and obtain the uniqueness of the inverse problem based on the idea of complex geometrical optics solutions. Then we give approximations to the conductivity coefficient by using the Fourier truncation regularization method and mollification method respectively. Under an a priori assump-tion of the conductivity, we estimate the errors between the conductivity coefficient and its approximation.Chapter 4 mainly studies an inverse problem of determining the unknown source ter-m only depending on spatial variable in the space fractional diffusion equation (?)tu(x, t)= d(?)xαu(x,t)+f(x),x∈R,t>0, where 1< α< 2 and this problem is ill-posed. Regular-ization solutions are obtained by employing three types of spectral regularization method under an a priori assumption. The detailed error estimates are also strictly established between exact solution and regularized solutions. In the numerical validation part, a di-rect problem is figured out by employing explicit difference method in order to obtain extra measured data. Finally, the regularization methods adopted by this chapter are val-idated by several numerical examples, which are not only effective for normal Gaussian distribution function, but also applied to the oscillation source function as well as the heat function with singularities in the complex plane, thus the feasibility and efficiency of the methods are validated.