A finite difference scheme for solving the undamped Timoshenko beam equations with both ends free

In this paper, a boundary feedback system of a class of non-uniform undamped Timoshenko beam with both ends free is considered. A linearized three-level difference scheme for the Timoshenko beam equations is derived by the method of reduction of order on uniform meshes. The unique solvability, unconditional stability and convergence of the difference scheme are proved by the discrete energy method. The convergence order in maximum norm is of order two in both space and time. The validity of this theoretical analysis is verified experimentally.