scholarly journals Optimal Polynomial Decay to Coupled Wave Equations and Its Numerical Properties

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
R. F. C. Lobato ◽  
S. M. S. Cordeiro ◽  
M. L. Santos ◽  
D. S. Almeida Júnior
Keyword(s):  
Decay Rate ◽  
Spectral Methods ◽  
Finite Differences ◽  
Wave Equations ◽  
Coupled System ◽  
Polynomial Decay ◽  
One Dimensional ◽  
Coupled Wave Equations ◽  
Optimal Polynomial ◽  
The One

In this work we consider a coupled system of two weakly dissipative wave equations. We show that the solution of this system decays polynomially and the decay rate is optimal. Computational experiments are conducted in the one-dimensional case in order to show that the energies properties are preserved. In particular, we use finite differences and also spectral methods.

scholarly journals A Novel Decay Rate for a Coupled System of Nonlinear Viscoelastic Wave Equations

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 203 ◽  
Author(s):  
Khaled Zennir ◽  
Sultan S. Alodhaibi
Keyword(s):  
Decay Rate ◽  
Wave Equations ◽  
Local Existence ◽  
Coupled System ◽  
Initial Energy ◽  
Energy Estimates ◽  
Contraction Mapping Theorem ◽  
Strong Damping ◽  
Nonlinear Viscoelastic ◽  
Viscoelastic Wave

The main goal of the present paper is to study the existence, uniqueness and behavior of a solution for a coupled system of nonlinear viscoelastic wave equations with the presence of weak and strong damping terms. Owing to the Faedo-Galerkin method combined with the contraction mapping theorem, we established a local existence in [ 0 , T ] . The local solution was made global in time by using appropriate a priori energy estimates. The key to obtaining a novel decay rate is the convexity of the function χ , under the special condition of the initial energy E ( 0 ) . The condition of the weights of weak and strong damping has a fundamental role in the proof. The existence of both three different damping mechanisms and strong nonlinear sources make the paper very interesting from a mathematics point of view, especially when it comes to unbounded spaces such as R n .

Including lateral velocity variations into true-amplitude common-shot wave-equation migration

Geophysics ◽  
2010 ◽  
Vol 75 (5) ◽  
pp. S175-S186 ◽  
Author(s):  
Daniela Amazonas ◽  
Rafael Aleixo ◽  
Gabriela Melo ◽  
Jörg Schleicher ◽  
Amélia Novais ◽  
...  
Keyword(s):  
Wave Equations ◽  
Heterogeneous Media ◽  
Synthetic Data ◽  
Approximate Solutions ◽  
Vertical Gradient ◽  
Correct Description ◽  
Lateral Velocity ◽  
One Dimensional ◽  
Velocity Variations ◽  
The One

In heterogeneous media, standard one-way wave equations describe only the kinematic part of one-way wave propagation correctly. For a correct description of amplitudes, the one-way wave equations must be modified. In media with vertical velocity variations only, the resulting true-amplitude one-way wave equations can be solved analytically. In media with lateral velocity variations, these equations are much harder to solve and require sophisticated numerical techniques. We present an approach to circumvent these problems by implementing approximate solutions based on the one-dimensional analytic amplitude modifications. We use these approximations to show how to modify conventional migration methods such as split-step and Fourier finite-difference migrations in such a way that they more accurately handle migration amplitudes. Simple synthetic data examples in media with a constant vertical gradient demonstrate that the correction achieves the recovery of true migration amplitudes. Applications to the SEG/EAGE salt model and the Marmousi data show that the technique improves amplitude recovery in the migrated images in more realistic situations.

scholarly journals A general renormalization procedure on the one-dimensional lattice and decay of correlations

2019 ◽  
Vol 19 (01) ◽  
pp. 1950003
Author(s):  
Artur O. Lopes
Keyword(s):  
Fixed Point ◽  
Final Section ◽  
Polynomial Decay ◽  
Renormalization Procedure ◽  
Real Numbers ◽  
One Dimensional ◽  
Point Potential ◽  
Renormalization Operator ◽  
The One ◽  
Natural Way

We present a general form of renormalization operator [Formula: see text] acting on potentials [Formula: see text]. We exhibit the analytical expression of the fixed point potential [Formula: see text] for such operator [Formula: see text]. This potential can be expressed in a natural way in terms of a certain integral over the Hausdorff probability on a Cantor type set on the interval [0,1]. This result generalizes a previous one by Baraviera, Leplaideur and Lopes where the fixed point potential [Formula: see text] was of Hofbauer type. For the potentials of Hofbauer type (a well-known case of phase transition) the decay is like [Formula: see text], [Formula: see text]. Among other things we present the estimation of the decay of correlation of the equilibrium probability associated to the fixed potential [Formula: see text] of our general renormalization procedure. In some cases we get polynomial decay like [Formula: see text], [Formula: see text], and in others a decay faster than [Formula: see text], when [Formula: see text]. The potentials [Formula: see text] we consider here are elements of the so-called family of Walters’ potentials on [Formula: see text] which generalizes a family of potentials considered initially by Hofbauer. For these potentials some explicit expressions for the eigenfunctions are known. In a final section we also show that given any choice [Formula: see text] of real numbers varying with [Formula: see text] there exists a potential [Formula: see text] on the class defined by Walters which has a invariant probability with such numbers as the coefficients of correlation (for a certain explicit observable function).

scholarly journals Numerical Study of the One-Dimensional Spin-Orbit Coupled System with SU(2) ⊗SU(2) Symmetry

2000 ◽  
Vol 69 (1) ◽  
pp. 242-247 ◽  
Author(s):  
Yasufumi Yamashita ◽  
Naokazu Shibata ◽  
Kazuo Ueda
Keyword(s):  
Numerical Study ◽  
Coupled System ◽  
Spin Orbit ◽  
One Dimensional ◽  
The One

scholarly journals Optimal polynomial decay for a coupled system of wave with past history

2020 ◽  
Vol 4 (1) ◽  
pp. 49-59 ◽  
Author(s):  
S. M. S. Cordeiro ◽  
◽  
R. F. C. Lobato ◽  
C. A. Raposo ◽  
◽  
...  
Keyword(s):  
Past History ◽  
Coupled System ◽  
Polynomial Decay ◽  
Optimal Polynomial
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