Identification of the zeroth-order coefficient in a time fractional diffusion equation

This paper is devoted to identify the zeroth-order coefficient in a time-fractional diffusion equation from two boundary measurement data in one-dimensional case. The existence and uniqueness of two kinds of weak solutions for the direct problem with Neumann boundary condition are proved. We provide the uniqueness for recovering the zeroth-order coefficient and fractional order simultaneously by the Laplace transformation and Gel'fand–Levitan theory. The identification of the zeroth-order coefficient is formulated into a variational problem by the Tikhonov regularization. The existence, stability and convergence of the solution for the variational problem are provided. We deduce an adjoint problem and then use a conjugate gradient method to solve the variational problem. Two numerical examples are provided to show the effectiveness of the proposed method.