In this thesis, we study the fully discrete local discontinuous Galerkin method for some time-fractional fourth-order differential equations. The method is based on a finite difference in time and local discontinuous Galerkin (LDG) methods in space. This scheme was employed to solve the problem and shown to converge with order O(τ −αh k+1 + τ 2−α + τ − α 2 h k+ 1 2 + h k+1). We find that the LDG method can actually achieve convergence rate equals O(τ 2−α + h k+1). The claims are justified accurately by numerical tests.