Wave equation with porous nonlinear acoustic boundary conditions generates a well-posed dynamical system

We consider a structural acoustic wave equation with nonlinear acoustic boundary conditions. This is a coupled system of second and first order in time partial differential equations, with boundary conditions on the interface. We prove wellposedness in the Hadamard sense for strong and weak solutions. The main tool used in the proof is the theory of nonlinear semigroups. We present the system of partial differential equations as a suitable Cauchy problem . Though the operator A is not maximally dissipative we are able to show that it is a translate of a maximally dissipative operator. The obtained semigroup solution is shown to satisfy a suitable variational equality, thus giving weak solutions to the system of PDEs. The results obtained (i) dispel the notion that the model does not generate semigroup solutions, (ii) provide treatment of nonlinear models, and (iii) provide existence of a correct state space which is invariant under the flow-thus showing that physical model under consideration is a dynamical system. The latter is obtained by eliminating compatibility conditions which have been assumed in previous work (on the linear case).     

Fourth-order problems with fully nonlinear boundary conditions

Differential inequality method, bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth-order differential equations under nonlinear boundary conditions, and many existing results are complemented.     

Parameter estimation in nonlinear evolution equations

We develop a convergence theory for identifying parameters in a general class of nonau-

tonomous nonlinear evolution equations. An application of this theory to a nonlinear reaction diffusion equation is presented.     

A Caginalp phase-field system with a nonlinear coupling

Our aim in this article is to study a Caginalp phase-field system based on a thermomechanical theory of deformable continua proposed by Green and Naghdi (called type III thermoelasticity) and with a nonlinear coupling. In particular, we prove the existence and uniqueness of solutions and the dissipativity of the associated semigroup. We then study the spatial behavior of the solutions in a semi-infinite cylinder, when such solutions exist.     

Parameter estimation in nonlinear stochastic differential equations

We discuss the problem of parameter estimation in nonlinear stochastic differential equations (SDEs) based on sampled time series. A central message from the theory of integrating SDEs is that there exist in general two time scales, i.e. that of integrating these equations and that of sampling. We argue that therefore, maximum likelihood estimation is computationally extremely expensive. We discuss the relation between maximum likelihood and quasi maximum likelihood estimation. In a simulation study, we compare the quasi maximum likelihood method with an approach for parameter estimation in nonlinear SDEs that disregards the existence of the two time scales.     

Parameter estimation in systems of nonlinear equations

Summary. A numerical method is presented to determine parameters in a system of non\-linear equations in the following sense: Proceeding from given experimental data, e.g., observation times and measurements, the minimum least-squares distance of the measured data from a fitting criterion depending on the solution of a system of nonlinear equations is to be computed. Specifically coupled mass equilibrium models are described in more detail that are used in pharmaceutical applications for receptor-ligand binding studies. They are used for instance for the radioimmunological determination of Fenoterol or related substances. Numerical results based on laboratory data are included to test the robustness of the algorithm implemented. Received May 24, 1993/Revised version received February 13, 1994     

Parameter estimation in nonlinear age-dependent population dynamics

This paper studies a parameter estimation problem for the Gurtin-MacCamyequation, which is a nonlinear model of age-dependent populationdynamics. In estimating parameters such as the death rate andthe fertility which depend on the age and the total populationfrom a set of fully discrete observations of the population,we use a backward finite-difference scheme. The function-spaceparameter estimation convergence (FSPEC) of this scheme is provedand numerical simulations are performed.     

Uniform stabilization of a nonlinear structural acoustic model with a Timoshenko beam interface

This paper is concerned with a nonlinear model which describes the interaction of sound and elastic waves in a two‐dimensional acoustic chamber in which one flat ‘wall’, the interface, is flexible. The composite dynamics of the structural acoustic model is described by the linearized equations for a gas defined on the interior of the chamber and the nonlinear Timoshenko beam equations on the interface. Uniform stability of the energy associated with the interactive system of partial differential equations is achieved by incorporating a nonlinear feedback boundary damping scheme in the equations for the gas and the beam. Copyright © 2006 John Wiley & Sons, Ltd.     

Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions

We prove an existence and location result for the third order functional nonlinear boundary value problem

     

Well-posedness for a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions

We consider a variable-coefficient wave equation with nonlinear damped acoustic boundary conditions. Well-posedness in the Hadamard sense for strong and weak solutions is proved by using the theory of nonlinear semigroups.     

Robust estimation of nonlinear structural models

The problem is considered of identification of nonlinear error-in-variables models under conditions of a heavy contamination of the sample, including the presence of outliers (i.e., anomalous observations). On the basis of robust estimation methods, we propose a development of the algorithms of adjusted and total least squares. This has enabled us to improve the accuracy of the response prediction in the presence of outliers in the sample. The algorithms are used for constructing the Engel curve from the budget survey data. In result, we draw better conclusions about the regularities of the household behavior when the income changes.     

Bifurcations of a micro-electromechanical nonlinear coupling system

The dynamical behavior of a micro-electromechanical nonlinear coupling system – deformable micromirror device, is investigated in this paper. In the literature some nonlinear phenomena have been explored by using the numerical method, and saddle-node bifurcation and periodic motions were discovered numerically. Overcoming the obstacle of the unsolvable of the equilibrium points, we analytically obtain the number and stability of the equilibrium points of the system discussed. The saddle-node bifurcation is obtained through the analytic method. Further, both codimension two bifurcations are revealed by the rigorous analysis. Finally, numerical simulations are in good agreement with the theoretical analysis.     

Phase-field systems with nonlinear coupling and dynamic boundary conditions

We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium.     

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

We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping, we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to zero, which provides rigorous justification for studying the case where the mass is equal to zero.     

Well-posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

In this paper, we prove the well-posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction $1\le n\le 4$. The method we use is based on a finite-dimensional approach by combining the Faedo-Galerkin method with suitable energy estimates and multiplier techniques.     

Parameter estimation of varying coefficients structural EV model with time series

In this paper, the parameters of a p-dimensional linear structural EV (error-in-variable) model are estimated when the coefficients vary with a real variable and the model error is time series. The adjust weighted least squares (AWLS) method is used to estimate the parameters. It is shown that the estimators are weakly consistent and asymptotically normal, and the optimal convergence rate is also obtained. Simulations study are undertaken to illustrate our AWLSEs have good performance.