# Hyperbolic Equations (hyperbolic + equation)

Distribution by Scientific Domains

## Selected Abstracts

### Green's function interpolations for prestack imaging

GEOPHYSICAL PROSPECTING, Issue 1 2000
Manuela Mendes
A new interpolation method is presented to estimate the Green's function values, taking into account the migration/inversion accuracy requirements and the trade-off between resolution and computing costs. The fundamental tool used for this technique is the Dix hyperbolic equation (DHE). The procedure, when applied to evaluate the Green's function for a real source position, uses the DHE to derive the root-mean-square velocity, vRMS, from the precomputed traveltimes for the nearest virtual sources, and by linear interpolation generates vRMS for the real source. Then, by applying the DHE again, the required traveltimes and geometrical spreading can be estimated. The inversion of synthetic data demonstrates that the new interpolation yields excellent results which give a better qualitative and quantitative resolution of the imaging sections, compared with those carried out by conventional linear interpolation. Furthermore, the application to synthetic and real data demonstrates the ability of the technique to interpolate Green's functions from widely spaced virtual sources. Thus the proposed interpolation, besides improving the imaging results, also reduces the overall CPU time and the hard disk space required, hence decreasing the computational effort of the imaging algorithms. [source]

### L1 Decay estimates for dissipative wave equations

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 5 2001
Albert Milani
Let u and v be, respectively, the solutions to the Cauchy problems for the dissipative wave equation $$u_{tt}+u_t-\Delta u=0$$\nopagenumbers\end(1) and the heat equation $$v_t-\Delta v=0$$\nopagenumbers\end(2) We show that, as $t\rightarrow+\infty$\nopagenumbers\end, the norms $\|\partial_t^k\,D_x^\alpha u(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end and $\|\partial_t^k\,D_x^\alpha v(\,\cdot\,,t)\|_{L^1({\rm R}^n)}$\nopagenumbers\end decay to 0 with the same polynomial rate. This result, which is well known for decay rates in $L^p({\rm R}^n)$\nopagenumbers\end with $2\leq p\leq+\infty$\nopagenumbers\end, provides another illustration of the asymptotically parabolic nature of the hyperbolic equation (1). Copyright © 2001 John Wiley & Sons, Ltd. [source]

### Local energy decay for a class of hyperbolic equations with constant coefficients near infinity

MATHEMATISCHE NACHRICHTEN, Issue 5 2010
Shintaro Aikawa
Abstract A uniform local energy decay result is derived to a compactly perturbed hyperbolic equation with spatial vari¬able coefficients. We shall deal with this equation in an N -dimensional exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data and the equation includes anisotropic variable coefficients {ai(x): i = 1, 2, ,, N }, which are not necessarily equal to each other (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials

MATHEMATISCHE NACHRICHTEN, Issue 13-14 2007
Maurizio Grasselli
Abstract In this article, we study the long time behavior of a parabolic-hyperbolic system arising from the theory of phase transitions. This system consists of a parabolic equation governing the (relative) temperature which is nonlinearly coupled with a weakly damped semilinear hyperbolic equation ruling the evolution of the order parameter. The latter is a singular perturbation through an inertial term of the parabolic Allen,Cahn equation and it is characterized by the presence of a singular potential, e.g., of logarithmic type, instead of the classical double-well potential. We first prove the existence and uniqueness of strong solutions when the inertial coefficient , is small enough. Then, we construct a robust family of exponential attractors (as , goes to 0). (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On the numerical solution of hyperbolic PDEs with variable space operator

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2009
Allaberen Ashyralyev
Abstract The first and second order of accuracy in time and second order of accuracy in the space variables difference schemes for the numerical solution of the initial-boundary value problem for the multidimensional hyperbolic equation with dependent coefficients are considered. Stability estimates for the solution of these difference schemes and for the first and second order difference derivatives are obtained. Numerical methods are proposed for solving the one-dimensional hyperbolic partial differential equation. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009 [source]

### Numerical solution of the one-dimensional wave equation with an integral condition

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2007
Abstract The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this research a numerical technique is developed for the one-dimensional hyperbolic equation that combine classical and integral boundary conditions. The proposed method is based on shifted Legendre tau technique. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 282,292, 2007 [source]

### Linear stability analysis and fourth-order approximations at first time level for the two space dimensional mildly quasi-linear hyperbolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001
R. K. Mohanty
Abstract In 1996, Mohanty et al. [1] presented a fourth-order finite difference solution of a two space dimensional nonlinear hyperbolic equation with Dirichlet boundary conditions. In 1998, Mohanty et al. [2] discussed a fourth-order approximation at first time level for the numerical solution of the one space dimensional hyperbolic equation. In both the cases, they have discussed the stability analysis for the linear hyperbolic equation having first-order space derivative terms. Recently, Mohanty et al. [3] have developed fourth-order difference formulas for the three space dimensional quasi-linear hyperbolic equations and obtained fourth-order approximation at first time level. In this article, we extend our strategy for solving the two space dimensional quasi-linear hyperbolic equation. An operator splitting method for a linear hyperbolic equation having a time derivative term is proposed. Linear stability analysis and fourth-order approximation at first time level for the two space dimensional quasi-linear hyperbolic equation are also discussed. The results of the numerical experiments are compared with the exact solution. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 607,618, 2001 [source]

### Arbitrary discontinuities in space,time finite elements by level sets and X-FEM

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 15 2004
Jack Chessa
Abstract An enriched finite element method with arbitrary discontinuities in space,time is presented. The discontinuities are treated by the extended finite element method (X-FEM), which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper-surface which is described by level sets. A space,time weak form for conservation laws is developed where the Rankine,Hugoniot jump conditions are natural conditions of the weak form. The method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non-linear Burgers' equations. By capturing the discontinuity in time as well as space, results are improved over capturing the discontinuity in space alone and the method is remarkably accurate. Implications to standard semi-discretization X-FEM formulations are also discussed. Copyright © 2004 John Wiley & Sons, Ltd. [source]

### Does the loss of regularity really appear?

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 10 2009
Xiaojun Lu
Abstract In the theory of weakly hyperbolic equations we have the effect of loss of regularity. The present paper devotes to the study of two problems. On the one hand we describe families of weakly hyperbolic Cauchy problems for which we have no loss of regularity. On the other hand we discuss the question if the loss of derivatives really appears. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Loss of regularity for the solutions to hyperbolic equations with non-regular coefficients,an application to Kirchhoff equation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2003
Fumihiko Hirosawa
We consider the Cauchy problem for second-order strictly hyperbolic equations with time-depending non-regular coefficients. There is a possibility that singular coefficients make a regularity loss for the solution. The main purpose of this paper is to derive an optimal singularity for the coefficient that the Cauchy problem is C, well-posed. Moreover, we will apply such a result to the estimate of the existence time of the solution for Kirchhoff equation. Copyright © 2003 John Wiley & Sons, Ltd. [source]

### Local energy decay for a class of hyperbolic equations with constant coefficients near infinity

MATHEMATISCHE NACHRICHTEN, Issue 5 2010
Shintaro Aikawa
Abstract A uniform local energy decay result is derived to a compactly perturbed hyperbolic equation with spatial vari¬able coefficients. We shall deal with this equation in an N -dimensional exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data and the equation includes anisotropic variable coefficients {ai(x): i = 1, 2, ,, N }, which are not necessarily equal to each other (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On the wellposedness of the Cauchy problem for weakly hyperbolic equations of higher order

MATHEMATISCHE NACHRICHTEN, Issue 10 2005
Piero D'Ancona
Abstract We study the wellposedness in the Gevrey classes Gs and in C, of the Cauchy problem for weakly hyperbolic equations of higher order. In this paper we shall give a new approach to the case that the characteristic roots oscillate rapidly and vanish at an infinite number of points. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### On the Cauchy problem for second order strictly hyperbolic equations with non,regular coefficients

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
Fumihiko Hirosawa
Abstract In this paper we shall consider some necessary and sufficient conditions for well,posedness of second order hyperbolic equations with non,regular coefficients with respect to time. We will derive some optimal regularities for well,posedness from the intensity of singularity to the coefficients by WKB representation of the solution and some counter examples which are constructed by using ideas of Floquet theory. [source]

### Estimates of hyperbolic equations in Hardy spaces

MATHEMATISCHE NACHRICHTEN, Issue 1 2003
Chen Chang
Abstract The aim of this paper is to study estimates of hyperbolic equations in Hardy classes. Consider the Cauchy problem P(Dt,Dx)u(t, x) = 0 for x , ,d and t > 0 with the initial conditions Djtu(0, x) = gj (x), j = 0, 1, ,, m , 1. We assume that the symbol ,,(,, ,) of P(Dt,Dx) can be factorized as ,,(,, ,) = (,,,j(,)) where ,j (,) = , j = 1, ,, m. We assume further that gj , Hpk (,d) for j = 1, ,, m , 1. Then the solution u of the problem (3.13) is in Hp(,d) provided k , (d, 1) and < p < ,. Here n = max{n1, ,, nm}. In particular, P(Dt, Dx)u = , ,u = 0 with u(0, x) = f(x) and (0, x) = g(x), then the solution u of the wave equation is in Hp(,d) provided k , (d , 1) and 0 < p < ,. [source]

### Linear stability analysis and fourth-order approximations at first time level for the two space dimensional mildly quasi-linear hyperbolic equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2001
R. K. Mohanty
Abstract In 1996, Mohanty et al. [1] presented a fourth-order finite difference solution of a two space dimensional nonlinear hyperbolic equation with Dirichlet boundary conditions. In 1998, Mohanty et al. [2] discussed a fourth-order approximation at first time level for the numerical solution of the one space dimensional hyperbolic equation. In both the cases, they have discussed the stability analysis for the linear hyperbolic equation having first-order space derivative terms. Recently, Mohanty et al. [3] have developed fourth-order difference formulas for the three space dimensional quasi-linear hyperbolic equations and obtained fourth-order approximation at first time level. In this article, we extend our strategy for solving the two space dimensional quasi-linear hyperbolic equation. An operator splitting method for a linear hyperbolic equation having a time derivative term is proposed. Linear stability analysis and fourth-order approximation at first time level for the two space dimensional quasi-linear hyperbolic equation are also discussed. The results of the numerical experiments are compared with the exact solution. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 607,618, 2001 [source]