The asymptotic stability of the vertical vibrations of a suspension bridge and the main cables from which the bridge is suspended by the cables is investigated. The history of the materials of the bridge and the effect of heat governed by Gurtin-Pipkin's thermal law on main cables are taking into consideration. We show that the damping induced by the infinite memory of the bridge and the heat on the main cables are strong enough to stabilize the system.

In this paper, we study the long-time behavior of a weakly dissipative viscoelastic equation with variable exponent nonlinearity of the form utt+Δ2u−∫0tg(t−s)Δu(s)ds+a|ut|n(·)−2ut−Δut=0, where n(.) is a continuous function satisfying some assumptions and g is a general relaxation function such that g′(t)≤−ξ(t)G(g(t)), where ξ and G are functions satisfying some specific properties that will be mentioned in the paper. Depending on the nature of the decay rate of g and the variable exponent n(.), we establish explicit and general decay results of the energy functional. We give some numerical illustrations to support our theoretical results. Our results improve some earlier works in the literature.

We consider a Rao-Nakra beam model with memory and thermal dissipation governed by Fourier's law of heat conduction. We show that the damping mechanism effected by the memory term on one of the wave equations corresponding to the bottom layer is enough to guarantee general stability of the Roa-Nakra beam equation. Thus, exponential and polynomial decay are special cases.

In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory.

In this paper, we consider a plate equation as a model for a suspension bridge with a general nonlinear internal feedback and time-varying weight. Under some conditions on the feedback and the coefficient functions, we establish a general decay estimate for the associated energy functional, from which the exponential and the polynomial, are particular cases.

A two-level mixed finite element method is developed and analyzed to solve the Darcy--Forchheimer equation modeling non-Darcy flows in porous media. Instead of solving a large nonlinear system of equations on a fine mesh, the two-level method provides the flexibility of solving a small nonlinear system of equations on a coarse mesh with mesh size H followed by solving one linear system of equations on a fine mesh with mesh size h. In constructing the two-level algorithm, we introduce a small positive constant $\epsilon$ to the original nonlinear term of the Darcy--Forchheimer equation to avoid difficulties associated with nondifferentiability of the Euclidean norm at zero. The priori error estimates for the velocity and pressure gradient are obtained in $L^2$ and $L^{3/2}$ discrete norms, respectively. These estimates state that if piecewise constant velocities and piecewise continuous, linear pressures are used, then the coarse and fine meshes are related by $h = O(H^2)$. Numerical examples are provided to show the efficiency and accuracy of the method. Numerical results show that the two-level Galerkin mixed finite element method is computationally more cost-effective than the standard Galerkin mixed finite element and the rate of convergence also agrees with the theoretical results.

Under necessary compatibility condition, and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and stability of the solution of generalized Dary-Forchheimer model.

We considered mixed finite element discretization of generalized Darcy–Forchheimer model in a two or three dimensional porous domain, using a piecewise constant and continuous piecewise linear finite elements. The existence, uniqueness, stability and convergence of the discrete solution are established. The error estimates for the velocity and pressure in L m and L m + 1 m respectively are obtained for any m ∈ ( 1 , 2 ]. Numerical examples are also presented to confirm the rate of convergence.