The paper is concerned with the inverse problem of identifying a space dependent diffusion coefficient in the uni-dimensional time-fractional diffusion equation with smooth initial function and specified boundary conditions. An implicit difference scheme for solving the forward problem is introduced. The stability, convergence and the uniqueness of the solution are established. A regularization parameter depending on the number of iterations is chosen and an optimal perturbation algorithm for inversion process is proposed. Several examples of non-homogeneous diffusion coefficients are selected to show the utility of the scheme. The inversion results establish that the inversion algorithm is convergent in view of the finite-dimensional approximation. Several theorems are also proved.