A wave absorption control of beam near fixed boundary by finite difference (FD) approach is presented. The FD equation of motion of the node under the influence of the boundary condition is compensated to be the same as that of the inner node free from the boundary condition with respect to the propagation characteristic solution by control. Then, no wave reflection occurs ; the fixed boundary condition virtually disappears. The control strategy consists of a real-time convolution integral of the measured displacement and the transfer function that is obtained by the inverse Laplace transformation of the frequency domain control transfer function. We confirmed the effectiveness of the wave controller by the time domain simulation on both FD system and distributed one as well as the frequency domain response calculation. The FD characteristic solution is qualitatively different from the distributed one, which gives the real time controller without applying the curve-fitting technique on the frequency-domain control transfer function.