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Linear Wave Equation (linear + wave_equation)

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Engineering42%

Selected Abstracts

Numerical error patterns for a scheme with hermite interpolation for 1 + 1 linear wave equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2004
Zuojin Zhu
Abstract Numerical error patterns were presented when the fourth-order scheme based on Hermite interpolation was used to solve the 1 + 1 linear wave equation. Since most non-linear equations for real systems can be converted into linear forms by using proper transformations, this study certainly pertains its practical significance. The analytical solution was obtained under inhomogeneous initial and boundary conditions. It was found that not only the Hurst index of an error train at a given position but also its spatial distribution is dependent on the ratio of temporal to spatial intervals. The solution process with the fourth-order scheme based on Hermite interpolation diverges as the ratio is greater than unity. The results show that regular error pattern and smaller maxima of absolute values of numerical errors can be obtained when the ratio is set as unity; while chaotic phenomena for the numerical error propagation process can appear when the ratio is less than unity. It was found that it is better to choose the ratio as unity for the numerical solution of 1 + 1 linear wave equation with the scheme; while other selections for the ratio in the scheme can bring about chaotic patterns for the numerical errors. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Local energy decay for linear wave equations with non-compactly supported initial data

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2004
Ryo Ikehata
Abstract A local energy decay problem is studied to a typical linear wave equation in an exterior domain. For this purpose, we do not assume any compactness of the support on the initial data. This generalizes a previous famous result due to Morawetz (Comm. Pure Appl. Math. 1961; 14:561,568). In order to prove local energy decay we mainly apply two types of new ideas due to Ikehata,Matsuyama (Sci. Math. Japon. 2002; 55:33,42) and Todorova,Yordanov (J. Differential Equations 2001; 174:464). Copyright © 2004 John Wiley & Sons, Ltd. [source]

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2004
Taeko Yamazaki
Abstract We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd. [source]

A (3+1)-dimensional Painlevé integrable model obtained by deformation

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 2 2002
Jun Yu
Abstract To find some non-trivial higher-dimensional integrable models (especially in (3+1) dimensions) is one of the most important problems in non-linear physics. An efficient deformation method to obtain higher-dimensional integrable models is proposed. Starting from (2+1)-dimensional linear wave equation, a (3+1)-dimensional non-trivial non-linear equation is obtained by using a non-invertible deformation relation. Further, the Painlevé integrability of the resulting model is also proved. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Numerical error patterns for a scheme with hermite interpolation for 1 + 1 linear wave equations

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 5 2004
Zuojin Zhu
Abstract Numerical error patterns were presented when the fourth-order scheme based on Hermite interpolation was used to solve the 1 + 1 linear wave equation. Since most non-linear equations for real systems can be converted into linear forms by using proper transformations, this study certainly pertains its practical significance. The analytical solution was obtained under inhomogeneous initial and boundary conditions. It was found that not only the Hurst index of an error train at a given position but also its spatial distribution is dependent on the ratio of temporal to spatial intervals. The solution process with the fourth-order scheme based on Hermite interpolation diverges as the ratio is greater than unity. The results show that regular error pattern and smaller maxima of absolute values of numerical errors can be obtained when the ratio is set as unity; while chaotic phenomena for the numerical error propagation process can appear when the ratio is less than unity. It was found that it is better to choose the ratio as unity for the numerical solution of 1 + 1 linear wave equation with the scheme; while other selections for the ratio in the scheme can bring about chaotic patterns for the numerical errors. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Perfectly matched layers for transient elastodynamics of unbounded domains

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2004
Ushnish Basu
Abstract One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337,1375], the authors presented, inter alia, time-harmonic governing equations of PMLs for anti-plane and for plane-strain motion of (visco-) elastic media. This paper presents (a) corresponding time-domain, displacement-based governing equations of these PMLs and (b) displacement-based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti-plane PML is found to be symmetric, whereas that of the plane-strain PML is not. Numerical results are presented for the anti-plane motion of a semi-infinite layer on a rigid base, and for the classical soil,structure interaction problems of a rigid strip-footing on (i) a half-plane, (ii) a layer on a half-plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd. [source]

Local energy decay for linear wave equations with non-compactly supported initial data

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16 2004
Ryo Ikehata
Abstract A local energy decay problem is studied to a typical linear wave equation in an exterior domain. For this purpose, we do not assume any compactness of the support on the initial data. This generalizes a previous famous result due to Morawetz (Comm. Pure Appl. Math. 1961; 14:561,568). In order to prove local energy decay we mainly apply two types of new ideas due to Ikehata,Matsuyama (Sci. Math. Japon. 2002; 55:33,42) and Todorova,Yordanov (J. Differential Equations 2001; 174:464). Copyright © 2004 John Wiley & Sons, Ltd. [source]