Standard discontinuous Galerkin methods, based on piecewise polynomials of degree q=0,1, are considered for temporal semi-discretization for second-order hyperbolic equations. The main goal of this paper is to present a simple and straightforward a priori error analysis of optimal order with minimal regularity requirement on the solution. Uniform norm in time error estimates are also proved. To this end, energy identities and stability estimates of the discrete problem are proved for a slightly more general problem. These are used to prove optimal order a priori error estimates with minimal regularity requirement on the solution. The combination with the classic continuous Galerkin finite element discretization in space variable is used to formulate a full-discrete scheme. The a priori error analysis is presented. Numerical experiments are performed to verify the theoretical results. 

The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order p and prove convergence of order p, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.

This paper deals with the stabilization problem of an axially moving string with a tip mass attached at the free end and subject to an external disturbance. The disturbance here is not uniformly bounded, and it is assumed to be exponentially increasing. First, the tip mass equation is designed under a boundary controller. By using this equation, the active disturbance rejection control (ADRC) technique is applied to design a disturbance observer, and it is shown that the observer can be estimated exponentially. Then, the closed-loop system is formulated and the well-posedness of the model is proved in the framework of the semigroup theory. The stability of the closed-loop system is then proved by means of the multiplier technique, where the energy system converges to equilibrium with an exponential manner. The efficiency of the obtained results is verified through numerical simulations. 

This paper concerns with the stabilization of an axially moving beam by an adaptive boundary control. We prove existence and uniqueness of the solution by means of nonlinear semigroup theory. Moreover, we construct the control through a low-gain adaptive velocity feedback. We also prove that the designed control is able to stabilize exponentially the closed loop system. Some numerical simulations are given to illustrate the theoretical results.   

Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag--Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented.  

In this paper, we consider a system modeling an axially moving viscoelastic string under a spatiotemporally varying tension. A mechanism consisted of a hydraulic touch-roll actuator pointed at the right boundary to suppress the transverse vibrations. We adopt the multiplier method to design a boundary control law and to prove an exponential stability result. However, this result is obtained provided that the lower bound of the tension in the string is larger than its time derivative. The effectiveness of the proposed control law is demonstrated via simulations. 

In this work, we investigate the asymptotic behavior of solutions of a viscoelastic flexible marine riser with vessel dynamics. Under a suitable control applied at the top end of the riser, we establish explicit decay rates for a large class of relaxation functions. In particular, exponentially and polynomially (or power type) decaying functions are included in this class. Our method is based on the multiplier technique. Numerical simulations justifying the effectiveness of the proposed boundary control to suppress the vibrations of the flexible marine riser are provided.

We study a second order hyperbolic initial-boundary value partial differential equation (PDE) with memory that results in an integro-differential equation with a convolution kernel. The kernel is assumed to be either smooth or no worse than weakly singular, that arise for example, in linear and fractional order viscoelasticity. Existence and uniqueness of the spatial local and global Galerkin approximation of the problem is proved by means of Picard's iteration. Then, spatial finite element approximation of the problem is formulated, and optimal order a priori estimates are proved by the energy method. The required regularity of the solution, for the optimal order of convergence, is the same as minimum regularity of the solution for second order hyperbolic PDEs. Spatial rate of convergence of the finite element approximation is illustrated by a numerical example. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 548–563, 2016