We investigate the initial-boundary value problem for time- fractional diffusion equations, where the time differentiation consists of a finite summation of Caputo derivatives with decreasing orders and positive constant coefficients. Based on several new properties of multinomial Mittag-Leffler functions, we establish the uniqueness and stability of the solution with respect to initial value and source term, from which the continuous dependency of Lipschitz type upon various coefficients is also verified. Concerning the numerical simulation, we develop a standard Galerkin finite element method and a fully discrete scheme, and give error estimates. This is a joint work with Dr. Bangti Jin (UC Riverside), Prof. Raytcho Lazarov and Dr. Zhi Zhou (Texas A&M Univ.).