Finite difference method

Abstract: In the low-frequency limit, the displacement currents in the Maxwell equations can be neglected. However, for numerical simulations, a small displacement current should be present to achieve numerical stability. This requirement leads to a large range of propagation velocities with high velocities for the high frequencies and low velocities for the low frequencies. As a consequence, the number of time steps may become large. I show that it is possible to transform mathematically the original physical problem to one that has propagation velocities with less frequency dependence. Hence, the number of time steps necessary for a signal to travel a certain distance with the lowest velocity is significantly reduced. A typical example shows a reduction in computational time by a factor of 40. A comparison of the solutions from plane-layered modeling in the frequency and wavenumber domain and the proposed method shows good agreement between the two. The proposed method can also be used for other systems of diffusive equations.

Abstract: This paper focuses on a new finite-time convergence disturbance rejection control scheme design for a flexible Timoshenko manipulator subject to extraneous disturbances. To suppress the shear deformation and elastic oscillation, position the manipulator in a desired angle, and ensure the finitetime convergence of disturbances, we develop three disturbance observers ( DOs ) and boundary controllers. Under the derived DOs-based control schemes, the controlled system is guaranteed to be uniformly bounded stable and disturbance estimation errors converge to zero in a finite time. In the end, numerical simulations are established by finite difference methods to demonstrate the effectiveness of the devised scheme by selecting appropriate parameters.