Order Hyperbolic Equations Research Articles

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.

Error estimates for single step fully discrete approximations for nonlinear second order hyperbolic equations

Optimal order rates of convergence are proved for fully discrete approximations for nonlinear second order hyperbolic equations which are based on a finite element approximation in the space variables and a single step method in the time variable. The r.h.s. of the differential equation can depend on u, u 1or ∇ u and the coefficients can depend on u. This paper generalizes the work of Bales where the r.h.s. and coefficients were allowed to depend on u.

The cauchy problem for the equation describing acoustic waves in a medium with dispersion

The Cauchy problem is considered for the third order hyperbolic equation which describes the non-stationary wave processes in a medium with dispersion and absorption. The exact solution is found, and is estimated asymptotically with respect to the small parameter which characterizes the dispersion of the velocity of sound.

CARLEMAN ESTIMATES AND INVERSE PROBLEMS FOR SECOND ORDER HYPERBOLIC EQUATIONS

The author considers the problem of finding the time-independent coefficients of a second order hyperbolic equation from the Cauchy data at the initial moment and on a part of the lateral surface of a cylindrical domain. Estimates of Hormander's Carleman type are obtained. On the basis of these estimates the uniqueness of the extension of the Cauchy data is proved, as well as the uniqueness of recovering the time-independent coefficients of hyperbolic equations.Bibliography: 9 titles.

Mixed initial‐boundary value problems for second order hyperbolic equations

On considere l'equation A(x,D)u=f dans un cylindre Ω=(−∞,+∞)×G, ou x=(x 0 ,x 1 ,...,x n ), x 0 ∈(−∞,+∞), (x 1 ,...,x n )∈G, D=[i∂/∂x 0 ,...,i∂/∂x n ], G est un domaine de R n a frontiere lisse compacte ∂G, x 0 est la variable de temps, f=0 pour x 0 <0, x∈Ω

On some solutions of second order hyperbolic differential equations with constant coefficients

If we seek solutions of the hyperbolic differential equationwhich depend only on the variables i and , we see that these solutions must be even in r and satisfy the differential equationThe object of this paper is to show that some recent results in the fractional calculus can be used to prove the following theorem.

Riccati Equations for Hyperbolic Partial Differential Equations with $L_2 (0,T;L_2 (\Gamma ))$—Dirichlet Boundary Terms

This paper studies the quadratic optimal control problem for second order (linear) hyperbolic partial differential equations defined on a bounded domain $\Omega \subset R^n $ with boundary $\Gamma $. Both the finite interval case $[0,T]$, $T < \infty $, and the infinite interval case $T = \infty $ (regulator problem) are considered. The distinguishing feature of the paper, which differentiates it from previous (scarce!) literature on the subject, is that the controls are only $L_2 (0,T;L_2 (\Gamma ))$-functions which act in the Dirichlet B.C. and that the corresponding solutions are penalized in the $L_2 (0,T;L_2 (\Omega ))$-norm (smoother controls, particularly in space, were taken in the few previous works on this subject). The well-posedness of this formulation stems from recent results by the authors about regularity of second order hyperbolic mixed problems [L - T.1], [L - T.3]. Under minimal assumptions, the optimal control is synthesized, in a pointwise feedback form, through an operator which is s...

Stability and accuracy for implicit semidiscretizations of hyperbolic problems

In this paper we give bounds for the error constants of certain classes of stable implicit finite difference methods for first order hyperbolic equations in one space dimension. We consider classes of methods that user downwind ands upwind points in the explicit part andR downwind andS upwind points in the implicit part, respectively, and that are of optimal orderp=min (r+R+s+S, 2(r+R+1), 2(s+S)).

Higher Order Single Step Fully Discrete Approximations for Second Order Hyperbolic Equations with Time Dependent Coefficients

Error estimates are proved for finite element approximations to the solution of an initial boundary value problem for a second order hyperbolic partial differential equation with time dependent coefficients. Optimal order rates of convergence in $L^2 $ are shown for single step fully discrete schemes. The order $\nu $ of the single step schemes is greater than or equal to one and less than or equal to four. A class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained. These methods avoid the repeated work of new matrix factorizations at each time level.

Long time effects of nonlinearity in second order hyperbolic equations

Long time effects of nonlinearity in second order hyperbolic equations

A Parabolic-Hyperbolic Free Boundary Problem

Change of phase problems with space-dependent melting temperature are considered. A weak formulation is given, pointing out that relevant differences occur with respect to the case of constant melting temperature. Next, analyzing the case in which weak solutions are suitably smooth, it is found that (i) mushy regions may appear even if volumetric heat sources are absent; (ii) a differential system is satisfied, consisting of parabolic equations and first order hyperbolic equations coupled through free boundaries; (iii) the interface conditions have the form of “unilateral constraints”. Finally, a model problem is studied and the existence of a smooth solution is proved.

Note on propagation of classical waves: II

1. Main results and notations. We consider Cauchy's problem for the classical wave equation in :with initial conditions u(0, x) = uo(x) and ∂tu(0, x) = u1 (x). In (1·1) δ and m2 stand for the Laplacian in and a constant respectively. Note that a second order hyper-bolic differential equation with real constants can be reduced to (1·1) ([1], p. 183). Let uj (j = 0,1) be C∞ funotions on whose support is contained in the closed ball BR = {xεl; |x| ≥R} for some R &gt; 0. In this note we shall show that the solution u(t, x) of (1·1) possesses the following properties.

High order difference schemes with reduced dispersion for hyperbolic differential equations

We investigate difference schemes for systems of first order hyperbolic differential equations in two space dimensions, possessing the following characteristics: 1. The spatial discretizations are fourth order accurate. 2. The time discretization is of explicit Runge-Kutta type and is also fourth order accurate. 3. The scaled stability boundary is approximately 1 2 2 . 4. The weights in the space discretizations and the Runge-Kutta parameters can be adapted in order to reduce the dispersion of dominant Fourier components. This method is illustrated by applying it to the shallow water equations simulating the motion of water in a shallow sea due to tidal forces. Since in such problems the dominant frequencies in the solution are known in advance, the method can take fully advantage of the possibility to tune the various parameters to these dominant frequencies.

Inttial-boundary value problems for second order hyperbolic equations with general boundary conditions II

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A Survey of One-Step Splitting Methods for Semi-Discrete First Order Hyperbolic Partial Differential Equations

In this paper spliting methods are discussed for first order hyperbolic partial differential equations via the method of lines, and in particular the time integration will be discussed. A class of one-step integration formulas is defined, which is shown to contain several well-known spliting methods. For a number of methods stability results are given. In der vorliegenden Arbeit werden Splitting-Methoden und insbesondere die Integration über die Zeit mittels der Linienmethode für hyperbolische partielle Differenialgleichungen erster Ordnung diskutiert. Es wird eine Klasse von Ein-Schritt-Integrationsformeln definiert, für die gezeigt wird, daß sie verschiedene wohlbekannte Splitting-Methoden enthält. Für einige Methoden werden Stabilitätsresultate angegeben.

Semidiscrete and Single Step Fully Discrete Approximations for Second Order Hyperbolic Equations With Time-Dependent Coefficients

${L^2}$ norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time-dependent coefficients. Optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data. The initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved.

Supplement to Semidiscrete and Single Step Fully Discrete Approximations for Second Order Hyperbolic Equations With Time-Dependent Coefficients

Supplement to Semidiscrete and Single Step Fully Discrete Approximations for Second Order Hyperbolic Equations With Time-Dependent Coefficients

An attribute-analysis of memory retrieval in rats: Additivity of different cueing treatments in alleviating “spontaneous forgetting” after a long interval

Optimal control of multisectional tunnel furnaces is considered applying the Nash game approach, developed for large-scale cascade systems by Özgüner and Perkins (1978). Temperature in each section is modelled by a first order hyperbolic equation. The equations are coupled in a cascade through boundary conditions. Nash PI feedback control is synthesized solving succesively LQ problems. The gains are characterized by appropriate Riccati and linear integro-differential equations. Given periodical changes of charge weight, open-loop fuel rate prognoses are formulated using the frequency method. Orthogonal collocation is applied to get lumped approximation. Some numerical results are given.

Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients

L 2 {L^2} norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time-dependent coefficients. Optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data. The initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved.

Supplement to: “Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients”

Supplement to: “Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time-dependent coefficients”