Order Hyperbolic Equations Research Articles

Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

A boundary-only solution is presented for dynamic analysis of elastic membranes under large deflections. The solution procedure is based on the analog equation method (AEM). According to this method, the three coupled nonlinear second order hyperbolic partial differential equations in terms of displacements, which govern the response of the membrane, are replaced with three Poisson's quasi-static equations under fictitious time dependent sources. The fictitious sources are established using a BEM-based procedure and the displacements as well as the stress resultants at any point are evaluated from their integrals representations. Numerical examples are presented which illustrate the method.

A Priori Error Estimates for Mixed Finite Element Approximations of the Acoustic Wave Equation

In this paper we derive optimal {a priori} $L^{\infty}(L^2)$ error estimates for mixed finite element displacement formulations of the acoustic wave equation. The computational complexity of this approach is equivalent to the traditional mixed finite element formulations of the second order hyperbolic equations in which the primary unknowns are pressure and the gradient of pressure. However, the displacement formulations with the physical variables of interest, displacement and pressure, requires less regularity on the displacement.

Time discretization of vibro‐impact

We consider vibro–impact problems, i.e. mechanical systems with a finite number of degrees of freedom submitted to perfect unilateral constraints. The dynamics is basically described by a second–order measure differential inclusion for the unknown position completed with a constitutive impact law. Another formulation of the problem as a frictionless sweeping process is possible: the unknown velocity belongs to an appropriate functional space and satisfies a first order measure differential inclusion. The equivalence of these two formulations is studied. They lead to time–discretizations written in terms of positions or in terms of velocities, respectively. We present these different schemes and we compare them on the simple test–problem of a bouncing ball. We recall the convergence results in the single constraint case. Moreover, an example of implementation of the scheme derived from the basic description of the dynamics is presented. Finally, in the multi–constraint case, we point out some theoretical and computational difficulties.

THE GOURSAT PROBLEM FOR HYPERBOLIC LINEAR THIRD ORDER EQUATIONS

The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.

The Goursat problem for hyperbolic linear third order equations

Abstract The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.

Highly accurate method for solving initial boundary value problem for first order hyperbolic differential equations

Restrictive Taylor approximation is a new approach similar to Restrictive Pade' approximation mentioned in [2] is proposed. An explicit method for solving a first order wave equation is treated. The suggested method is of high accuracy whatever the exact solution is too large. It treats also the discontinuous behavior of the initial data for the function or its derivative. The stability condition and comparison between the considered method and Lax-Wendroff method are discussed.

Vessiot structure for manifolds of $(p,q)$-hyperbolic type: Darboux integrability and symmetry

It is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is Darboux integrable, then its local Cauchy problem may be solved by ordinary differential equations. In addition, such an equation has infinitely many non-trivial conservation laws. Moreover, Darboux integrable equations have properties in common with infinite dimensional completely integrable systems. In this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation, its hyperbolic structure, to study the Darboux integrability of the class E of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables. It is shown that the problem of classifying the Darboux integrable equations in E contains, as a subproblem, that of classifying the manifolds of (p, q)hyperbolic type of rank 4 and dimension 2k + 3, k ≥ 2; p = 2, q ≥ 2. In turn, it is shown that the problem of classifying these manifolds in the two (lowest) cases (p, q) = (2, 2), (2, 3) contains, as a subproblem, the classification problem for Lie groups. This generalizes classical results of E. Vessiot. The main result is that if an equation in E is (2,2)or (2,3)-Darboux integrable on the k-jets, k ≥ 2, then its intrinsic hyperbolic structure admits a Lie group of symmetries of dimension 2k− 1 or 2k− 2, respectively. It follows that part of the moduli space for the Darboux integrable equations in E is determined by isomorphism classes of Lie groups. The Lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic (or, Riemann) invariants of the equation, the tangential characteristic symmetries. The isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding Darboux integrable partial differential equation.

Asymptotic Behavior of the Energy of Vibration of a Moving String with Varying Lengths

In this paper, the author studies the initial-moving boundary value problem of a linear second- order hyperbolic equation describing the vibration of an axially moving string with time-varying lengths. The explicit expression of the solution is presented, and the stability of the system is investigated. It is shown that the amplitude of vibration is uniformly bounded for any given initial conditions; moreover, the energy of the system increases without bound with polynomial growth as the length of the string tends to zero and decreases to zero with polynomial decay when the length tends to infinity.

On third order hyperbolic equations

We shall consider the third order hyperbolic equation in [0,T] ×Rx \(\left\{ {_{u\left( {0,x} \right) = u_0 \left( x \right), ut\left( {0,x} \right) = u_1 \left( x \right), u_{tt} \left( {0,x} \right) = u_2 \left( x \right),}^{u_{ttt} - t^\alpha u_{txx} + t^\beta u_{ttx} + t^\eta u_{tx} + t^\sigma u_{tt} + t^\mu u_x + t^\omega u_t + t^\theta u = 0} } \right.\) where α≥ 2, Β ≥ 1, η ≥ 0, λ ≥ 0, Σ≥ 0, Μ ≥ 0, Ω ≥ 0 and θ≥ 0 are integers. We prove that the Cauchy problem (1) is Gevrey well-posed.

Shape Derivative in the Wave Equation with Dirichlet Boundary Conditions

The aim of this paper is to give a full analysis of the the shape differentiability for the solution to the second order hyperbolic equation with Dirichlet boundary conditions. The implicit function theorem does not work to solve the problem of weak regularity of the data; nevertheless by a more technical approach we prove an analogous result. We will first prove the theorem under strong regularity of the right hand side, then using the hidden regularity we will prove the shape derivative continues to exist under weak condition of regularity. We end up with a second order shape derivative for this problem.

Exact Neumann boundary controllability for second order hyperbolic equations

Using HUM, we study the problem of exact controllability with Neumann boundary conditions for second order hyperbolic equations. We prove that these systems are exactly controllable for all initial states in $L^2({\mit\Omega})\times (H^1({\mit\Omega}))'$

The effect of spatial quadrature on finite element galerkin approximations to hyperbolic integro-differential equations

The purpose of this paper is to study the effect of numerical quadrature on the finite element approximations to the solutions of hyperbolic intego-differential equations. Both semidiscrete and fully discrete schemes are analyzed and optimal estimates are derived in L ∞(H 1)L ∞(L 2) norms and quasi-optimal estimate in L ∞(L ∞) norm using energy arguments. Further, optimal L(L 2)-estimates are shown to hold with minimal smoothness assumptions on the initial functions. The analysis in the present paper not only improves upon the earlier results of Baker and Dougalis [SIAM J. Numer. Anal. 13 (1976), pp. 577-598] but also confirms the minimum smoothness assumptions of Rauch [SIAM J. Numer. Anal. 22 (1985), pp. 245-249] for purely second order hyperbolic equation with quadrature.

The Trotter-Kato theorem and approximation of PDEs

We present formulations of the Trotter-Kato theorem for approximation of linear C 0 {}_0 -semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes’ equation as illustrative examples.

Local and global results for wave maps I

We consider the initial value problem for wave-maps corresponding to constant coefficient second order hyperbolic equations in dimensions,. We prove that this problem is globally well-posed for initial data which is small in the homogeneous Besov space . Our second result deals with more regular solutions; it essentially says that if in addition the initial data is in then the solutions stay bounded in the same space. In part II of this work we shall prove that the same result holds in dimensions n = 2,3.

A one dimensional analogue of the vorticity equation

We study the qualitative properties of the one dimensional analogue of the Helmholtz vorticity advection equation. The second order hyperbolic equation has the unusual characteristic of disturbances propagating at infinite speed. The global solution for Goursat data is given in closed form. We also obtain qualitative results on the nodal curve where the solution is zero. A related perturbation problem is considered and solutions for small data are obtained. The forced vorticity equation admits a class of soliton solutions.

The optimal explicit unconditionally stable box scheme

The finite difference “box” scheme, (see also [1],[2]), is considered on the simplest possible model of single first order linear hyperbolic equation: ut+mux=0 with constant, coefficient m, and one space variable. The optimal version of the scheme, which is nonoscilating and unconditionally stable with respect to the initial and boundary conditions, is derived in the class of box schemes of the order at, least one. If apropriately iterated, this ścinane may be applied to general systems of quasilinear first order hyperbolic equations in one space variable, as an explicit, unconditionally stable solver. For more than one space variable this solver is applicable via splitting (see [3]).

ON THE ENERGY DECAY OF TWO COUPLED STRINGS THROUGH A JOINT DAMPER

The stabilization of two coupled strings with fixed boundaries attached to a linear damper at their joint of connection is studied. The transformation used in a previous approach to convert the second order hyperbolic equations into a first order system results in a loss of equivalence. It is shown that when the ratio of the wave speeds of the two strings is an irrational number, the system is asymptotically, but not exponentially, stable. The method is applied to the other cases with different joint and boundary conditions.

Almost global existence for solutions of a nonlinear fourth order hyperbolic equation

We study the long time existence of the solutions of the Cauchy problem for a class of nonlinear fourth order hyperbolic equations, in which the principal part is given by the composition of two waves operators with different propagation speeds. The presence of these two speeds makes this problem essentially different from the correspondent one for the second order wave equation. In fact, in the present case, we cannot apply the approach of S. Klainerman, based on the invariance of the D'Alembertian operator under the complete Lorentz group. Furthermore, even the alternative method used by F. John and S. Klainerman, only for the wave equation in three space dimensions, does not seem directly applicable. Nevertheless, thanks to the special structure of the equations, we are able to show the almost global existence of the solutions to the related Cauchy problem.

Approximation of H∞-control problem in hilbert spaces

We consider the approximation of a H∞-control problem, with boimded input and output. The approximating operators involved converge pointwise, except the disturbance operators Dh which converge in the uniform norm. We show that the existence of solution for the continuous H∞-control problem implies existence of solution for the approximating H∞ problem. We also prove pointwise convergence of the solutions to the Riccati equations associated to the approximating H∞ problem to the solution to the Riccati equations associated to the continuous problem. We show that the solution to the approximating H∞ problem solves in a weak sense the continuous problem. As an example we considered a control system governed by a first order hyperbolic equation.

An ill-posed problem of the continuation of transient data for a hyperbolic equation in a three-dimensional inhomogeneous half-space

A time-domain data continuation problem for a second order hyperbolic equation in a three-dimensional inhomogeneous half-space, which is a common problem in time-domain layer-stripping, is considered. Two different approaches, namely the local continuation approach and the wave-splitting approach, are used to propagate the surface data (the Dirichlet and Neumann data) to the data at deeper planes. The local continuation approach requires a regularization in order to obtain a stable numerical result. The wave-splitting approach uses larger transverse spatial domain of dependence, and gives reasonably stable results even without any additional smoothing process.