Variable-coefficient Wave Equation Research Articles

Stability for variable-coefficient wave equation with local frictional damping and memory type dynamic boundary conditions

Stability for variable-coefficient wave equation with local frictional damping and memory type dynamic boundary conditions

Exponential stability result for the wave equation with Kelvin–Voigt damping and past history subject to Wentzell boundary condition and delay term

In this paper, we present an analysis of stability of solutions corresponding to a variable coefficient's wave equation subject to a locally Kelvin–Voigt damping and distributed effect driven by a nonnegative function with dynamic Wentzell boundary conditions and delay term. By using frequency domain approach method, we show that under a suitable assumption between the internal damping function and the boundary delay feedback, considering some geometrical assumptions on the boundary of , supposing that the relaxation function decay exponentially to zero, that the energies of the problem decay exponentially to zero.

Existence and General Energy Decay of Solutions to a Coupled System of Quasi-Linear Viscoelastic Variable Coefficient Wave Equations with Nonlinear Source Terms

Viscoelastic damping phenomena are ubiquitous in diverse kinds of wave motions of nonlinear media. This arouses extensive interest in studying the existence, the finite time blow-up phenomenon and various large time behaviors of solutions to viscoelastic wave equations. In this paper, we are concerned with a class of variable coefficient coupled quasi-linear wave equations damped by viscoelasticity with a long-term memory fading at very general rates and possibly damped by friction but provoked by nonlinear interactions. We prove a local existence result for solutions to our concerned coupled model equations by applying the celebrated Faedo-Galerkin scheme. Based on the newly obtained local existence result, we prove that solutions would exist globally in time whenever their initial data satisfy certain conditions. In the end, we provide a criterion to guarantee that some of the global-in-time-existing solutions achieve energy decay at general rates uniquely determined by the fading rates of the memory. Compared with the existing results in the literature, our concerned model coupled wave equations are more general, and therefore our theoretical results have wider applicability. Modified energy functionals (can also be viewed as certain Lyapunov functionals) play key roles in proving our claimed general energy decay result in this paper.

Euler–Darboux–Poisson Equation in Context of the Traveling Waves in a Strongly Inhomogeneous Media

The existence of traveling waves in an inhomogeneous medium is a vital problem, the solution of which can help in modeling the wave propagation over long distances. Such waves can be storm waves or tsunami waves in the seas and oceans. The presence of solutions in the form of traveling waves indicates that the wave propagates without reflection and, therefore, can transfer energy over long distances. Traveling waves within the framework of the 1D variable-coefficient wave equation exist only for certain configurations of an inhomogeneous medium, some of which can be found by transforming the original equation to the Euler–Darboux–Poisson equation. The solution of the last equation for certain parameter values is expressed in elementary functions, which are the sum of waves running in opposite directions. The mathematical features of such a transformation are discussed in this paper.

Stability for semilinear wave equation of variable coefficients with acoustic boundary conditions and general boundary memory feedback

In this paper, we consider a semilinear wave equation with variable coefficients in a smooth domain, subject to acoustic boundary conditions and dissipative boundary memory feedback, where a general Borel measure is involved. We show that the decay rates of the energy associated to the semilinear systems are given implicitly as solutions of a first order nonlinear, dissipative ODE, which recovers and extends some of the results from the literature. To prove the main result we use energy multiplier methods, geometric analysis combined with Lasiecka and Tataru arguments.

On Construction and Properties of Compact 4th Order Finite-Difference Schemes for the Variable Coefficient Wave Equation

On Construction and Properties of Compact 4th Order Finite-Difference Schemes for the Variable Coefficient Wave Equation

Energy Decay for Wave Equation of Variable Coefficients with Dynamic Boundary Conditions and Time-Varying Delay

Energy Decay for Wave Equation of Variable Coefficients with Dynamic Boundary Conditions and Time-Varying Delay

Polynomial Decay Rate of a Variable Coefficient Wave Equation with Memory Type Acoustic Boundary Conditions

Polynomial Decay Rate of a Variable Coefficient Wave Equation with Memory Type Acoustic Boundary Conditions

Traveling Waves in Shallow Seas of Variable Depths

The problem of the existence of traveling waves in inhomogeneous fluid is very important for enabling an explanation of long-distance wave propagations such as tsunamis and storm waves. The present paper discusses new solutions to the variable-coefficient wave equations describing traveling waves in fluid layers of variable depths (1D shallow-water theory). Such solutions are obtained by using the transformation methods when variable-coefficient equations can be reduced to the constant coefficient equation when the existence of traveling waves is evident. It is shown that there is a wide class of monotonic bottom profiles (discrete set) that allow the existence of traveling waves that are not reflected in a strongly inhomogeneous water medium. Their temporal shape changes with distance, mainly near the water–land boundary (shoreline). Traveling waves can transfer the wave energy over a long distance that is often observed at the transoceanic propagation of tsunami waves.

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.

Periodic solutions of a semilinear variable coefficient wave equation under asymptotic nonresonance conditions

We consider the periodic solutions of a semilinear variable coefficient wave equation arising from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The variable coefficient characterizes the inhomogeneity of media and its presence usually leads to the destruction of the compactness of the inverse of the linear wave operator with periodic-Dirichlet boundary conditions on its range. In the pioneering work of Barbu and Pavel (1997), they gave the existence and regularity of the periodic solution for Lipschitz, nonresonant and monotone nonlinearity under the assumption ηu > 0 (see Section 2 for its definition) on the coefficient u(x) and left the case ηu = 0 as an open problem. In this paper, by developing the invariant subspace method and using the complete reduction technique and Leray-Schauder theory, we obtain the existence of periodic solutions for such a problem when the nonlinear term satisfies the asymptotic nonresonance conditions. Our result not only does not need any requirements on the coefficient except the natural positivity assumption (i.e., u(x) > 0) but also does not need the monotonicity assumption on the nonlinearity. In particular, when the nonlinear term is an odd function and satisfies the global nonresonance conditions, there is only one (trivial) solution to this problem on the invariant subspace.

Energy decay for a wave equation of variable coefficients with logarithmic nonlinearity source term

We investigate the variable-coefficient wave equation with nonlinear acoustic boundary conditions and logarithmic nonlinearity source term. In addition, the damping is nonlinear, and plays a role only in part of the boundary. Using the semigroup theory, we state the existence and uniqueness of the local solutions. Accordingly, we prove that under some assumptions on the initial data, the local solution can be extended to be global. Particularly, it has been shown that the decay rates of the system are given implicitly as solutions to a first-order ODE by applying the Riemannian geometry method.

Decay and blow-up for a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity

In this article, we consider a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity and dynamic boundary conditions in a bounded domain. The existence of a global solution is given by use of the potential well method. In the stable set, with some suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity by use of the Riemannian geometry method, logarithmic Sobolev inequality, and Lyapunov functional method. Meanwhile, in the unstable set, blow-up of the solution is also obtained.

On higher-order compact ADI schemes for the variable coefficient wave equation

We consider an initial-boundary value problem for the n-dimensional wave equation, n⩾2, with the variable sound speed with the nonhomogeneous Dirichlet boundary conditions. We construct and study three-level in time and compact in space three-point in each spatial direction alternating direction implicit (ADI) schemes having the approximation orders O(ht2+|h|4) and O(ht4+|h|4) on the uniform rectangular mesh. The study includes stability bounds in the strong and weak energy norms, the discrete energy conservation law and the error bound of the order O(ht2+|h|4) for the first scheme as well as a short justification of the approximation order O(ht4+|h|4) for the second scheme. We also present results of numerical experiments.

Uniform stabilization of a variable coefficient wave equation with nonlinear damping and acoustic boundary

ABSTRACT In this paper we consider uniform stabilization of the wave equation with variable coefficients in a bounded domain. The nonlinear damping is put partly on the interior of the domain and partly on the acoustic boundary. Under some checkable conditions on the coefficients, the energy decay results are established by Riemannian geometry method.

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.

Uniform Stabilization of 1-D Coupled Wave Equations with Anti-dampings and Joint Delayed Control

In this paper, we consider the uniform stabilization problem for a coupled 1-d variable coefficient wave equation with anti-dampings in the interior domain and joint difference-type control. We use the integral-type feedback control to stabilize the system, in which the integral kernel functions are regarded as parameters. Our goal in the present paper is to select appropriate kernel functions such that the closed-loop system is exponentially stable. We first choose an exponentially stable system as a target system. Meanwhile, we introduce an appropriate auxiliary system. After that we construct two transformations. The first one is used to establish the equivalence between the original controlled time-delay system and the auxiliary system. This step mainly removes the effect of input memory. The second one is used to establish the equivalence between the auxiliary system and the stable target system. This step is to eliminate the negative effect of the anti-damping. Herein we mainly give an approach to selecting parameter functions in both transformations and hence get an exponentially stable-type feedback controller. Finally, some simulations are used to show the effectiveness of the proposed controller.

Polynomial decay of a variable coefficient wave equation with an acoustic undamped boundary condition

We consider a variable coefficient wave equation with an acoustic undamped condition. This is a coupled system of second and first order in time partial differential equations, with an acoustic boundary condition on the interface. The Riemannian geometry method is applied to deal with the variable coefficients. Under some checkable conditions on the coefficients we obtain the polynomial energy decay.

The Conservative Time High-Order AVF Compact Finite Difference Schemes for Two-Dimensional Variable Coefficient Acoustic Wave Equations

In this paper, we develop and analyze the energy conservative time high-order AVF compact finite difference methods for variable coefficient acoustic wave equations in two dimensions. We first derive out an infinite-dimensional Hamiltonian system for the variable coefficient wave equations and apply the spatial fourth-order compact finite difference operator to the equations of the system to obtain a semi-discrete approximation system, which can be cast into a canonical finite-dimensional Hamiltonian form. We then apply the second-order and fourth-order AVF techniques to propose the fully discrete energy conservative time high-order AVF compact finite difference methods for wave equations in two dimensions. We prove that the proposed semi-discrete and fully-discrete schemes satisfy energy conservations in the discrete forms. We further prove that the semi-discrete scheme has the fourth-order convergence order in space and the fully-discrete AVF compact finite difference method has the fourth-order convergence order in both time and space. Numerical tests confirm the theoretical results.

Energy decay and global solutions for a damped free boundary fluid–elastic structure interface model with variable coefficients in elasticity

ABSTRACT We study a free boundary fluid-structure interaction model. In the model, a viscous incompressible fluid interacts with an elastic body via the common boundary. The motion of the fluid is governed by Navier–Stokes equations while the displacement of the elastic structure is described by variable coefficient wave equations. The dissipation is placed on the common boundary between the fluid and the elastic body. Given small initial data, the global existence of the solutions of this system is proved and the exponential decay of solutions is obtained.