scholarly journals An Operator Method for the Stability of Inhomogeneous Wave Equations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 324 ◽  
Author(s):  
Ginkyu Choi ◽  
Soon-Mo Jung ◽  
Jaiok Roh
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Operator Method ◽  
Ulam Stability ◽  
Inhomogeneous Wave ◽  
Partial Derivatives ◽  
The Stability

In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, u t t ( x , t ) − c 2 ▵ u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.

scholarly journals A Dilation Invariance Method and the Stability of Inhomogeneous Wave Equations

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 70 ◽  
Author(s):  
Ginkyu Choi ◽  
Soon-Mo Jung
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Ulam Stability ◽  
Inhomogeneous Wave ◽  
Partial Derivatives ◽  
Inhomogeneous Wave Equation ◽  
Time Variables ◽  
The Stability

We apply the method of a kind of dilation invariance to prove the generalized Hyers-Ulam stability of the (inhomogeneous) wave equation with a source, u t t ( x , t ) − c 2 ▵ u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives in each of spatial and the time variables.

scholarly journals Some arguments for the wave equation in Quantum theory

2021 ◽  
Vol 5 (1) ◽  
pp. 314-336
Author(s):  
Tristram de Piro ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Continuity Equation ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Atomic System ◽  
Balmer Series ◽  
Inertial Frames ◽  
The Stability

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

Fractional Laplacians viscoacoustic wavefield modeling with k-space-based time-stepping error compensating scheme

Geophysics ◽  
2020 ◽  
Vol 85 (1) ◽  
pp. T1-T13 ◽  
Author(s):  
Ning Wang ◽  
Tieyuan Zhu ◽  
Hui Zhou ◽  
Hanming Chen ◽  
Xuebin Zhao ◽  
...  
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Time Derivative ◽  
Time Stepping ◽  
Long Time ◽  
Sampling Intervals ◽  
Spatial Derivatives ◽  
The Stability ◽  
Wavefield Extrapolation ◽  
Space Approach

The spatial derivatives in decoupled fractional Laplacian (DFL) viscoacoustic and viscoelastic wave equations are the mixed-domain Laplacian operators. Using the approximation of the mixed-domain operators, the spatial derivatives can be calculated by using the Fourier pseudospectral (PS) method with barely spatial numerical dispersions, whereas the time derivative is often computed with the finite-difference (FD) method in second-order accuracy (referred to as the FD-PS scheme). The time-stepping errors caused by the FD discretization inevitably introduce the accumulative temporal dispersion during the wavefield extrapolation, especially for a long-time simulation. To eliminate the time-stepping errors, here, we adopted the [Formula: see text]-space concept in the numerical discretization of the DFL viscoacoustic wave equation. Different from existing [Formula: see text]-space methods, our [Formula: see text]-space method for DFL viscoacoustic wave equation contains two correction terms, which were designed to compensate for the time-stepping errors in the dispersion-dominated operator and loss-dominated operator, respectively. Using theoretical analyses and numerical experiments, we determine that our [Formula: see text]-space approach is superior to the traditional FD-PS scheme mainly in three aspects. First, our approach can effectively compensate for the time-stepping errors. Second, the stability condition is more relaxed, which makes the selection of sampling intervals more flexible. Finally, the [Formula: see text]-space approach allows us to conduct high-accuracy wavefield extrapolation with larger time steps. These features make our scheme suitable for seismic modeling and imaging problems.

Time‐domain behavior of wide‐angle one‐way wave equations

Geophysics ◽  
1991 ◽  
Vol 56 (3) ◽  
pp. 382-384
Author(s):  
A. H. Kamel
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Source Function ◽  
Plane Waves ◽  
Wide Angle ◽  
Wave Processes ◽  
Inhomogeneous Wave ◽  
Dirac Delta ◽  
Inhomogeneous Wave Equation ◽  
Standard Tool

The constant‐coefficient inhomogeneous wave equation reads [Formula: see text], Eq. (1) where t is the time; x, z are Cartesian coordinates; c is the sound speed; and δ(.) is the Dirac delta source function located at the origin. The solution to the wave equation could be synthesized in terms of plane waves traveling in all directions. In several applications it is desirable to replace equation (1) by a one‐way wave equation, an equation that allows wave processes in a 180‐degree range of angles only. This idea has become a standard tool in geophysics (Berkhout, 1981; Claerbout, 1985). A “wide‐angle” one‐way wave equation is designed to be accurate over nearly the whole 180‐degree range of permitted angles. Such formulas can be systematically constructed by drawing upon the connection with the mathematical field of approximation theory (Halpern and Trefethen, 1988).

scholarly journals Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

2012 ◽  
Vol 17 (2) ◽  
pp. 245
Author(s):  
N.H. Sweilam ◽  
T.A. Assiri
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Error Analysis ◽  
Wave Equations ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Explicit Finite Difference ◽  
The Stability

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented.   

Constraints on the anisotropic parameters for pseudoelastic vertical transverse isotropy wave equations and the applications on imaging carbonate reservoirs

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. C85-C94 ◽  
Author(s):  
Houzhu (James) Zhang ◽  
Hongwei Liu ◽  
Yang Zhao
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Wave Velocity ◽  
Wave Equations ◽  
Transverse Isotropy ◽  
S Wave ◽  
Elastic Wave Equation ◽  
Computation Efficiency ◽  
The Stability ◽  
Vertical Transverse Isotropy

Seismic anisotropy is an intrinsic elastic property. Appropriate accounting of anisotropy is critical for correct and accurate positioning seismic events in reverse time migration. Although the full elastic wave equation may serve as the ultimate solution for modeling and imaging, pseudoelastic and pseudoacoustic wave equations are more preferable due to their computation efficiency and simplicity in practice. The anisotropic parameters and their relations are not arbitrary because they are constrained by the energy principle. Based on the investigation of the stability condition of the pseudoelastic wave equations, we have developed a set of explicit formulations for determining the S-wave velocity from given Thomsen’s parameters [Formula: see text] and [Formula: see text] for vertical transverse isotropy and tilted transverse isotropy media. The estimated S-wave velocity ensures that the wave equations are stable and well-posed in the cases of [Formula: see text] and [Formula: see text]. In the case of [Formula: see text], a common situation in carbonate, a positive value of S-wave velocity is needed to avoid the wavefield instability. Comparing the stability constraints of the pseudoelastic- with the full-elastic wave equation, we conclude that the feasible range of [Formula: see text] and [Formula: see text] was slightly larger for the pseudoelastic assumption. The success of achieving high-accuracy images and high-quality angle gathers using the proposed constraints is demonstrated in a synthetic example and a field example from Saudi Arabia.

scholarly journals On the Stability of Wave Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Soon-Mo Jung
Keyword(s):  
Wave Equation ◽  
Ulam Stability ◽  
Differentiable Functions ◽  
The Stability

We prove the generalized Hyers-Ulam stability of the wave equation,Δu=(1/c2)utt, in a class of twice continuously differentiable functions under some conditions.

scholarly journals A Fixed Point Approach to the Stability of an Integral Equation Related to the Wave Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Soon-Mo Jung
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Wave Equation ◽  
Fixed Point Method ◽  
Point Method ◽  
Ulam Stability ◽  
Fixed Point Approach ◽  
The Stability

We will apply the fixed point method for proving the generalized Hyers-Ulam stability of the integral equation1/2c∫x-ctx+ctuτ,t0dτ=ux,twhich is strongly related to the wave equation.

scholarly journals Stability of the Wave Equation with a Source

2018 ◽  
Vol 2018 ◽  
pp. 1-4 ◽  
Author(s):  
Soon-Mo Jung ◽  
Seungwook Min
Keyword(s):  
Wave Equation ◽  
Ulam Stability ◽  
Partial Derivatives

We prove the generalized Hyers-Ulam stability of the wave equation with a source, uttx,t-c2uxxx,t=fx,t, for a class of real-valued functions with continuous second partial derivatives in x and t.

Stability and regularity transmission for coupled beam and wave equations through boundary weak connections

2020 ◽  
Vol 26 ◽  
pp. 73
Author(s):  
Bao-Zhu Guo ◽  
Han-Jing Ren
Keyword(s):  
Wave Equation ◽  
Real Axis ◽  
Wave Equations ◽  
Basis Property ◽  
Coupled System ◽  
Finite Point ◽  
The Real ◽  
Riesz Basis Property ◽  
Asymptotic Expressions ◽  
The Stability

In this paper, we consider stability for a hyperbolic-hyperbolic coupled system consisting of Euler-Bernoulli beam and wave equations, where the structural damping of the wave equation is taken into account. The coupling is actuated through boundary weak connection in the sense that after differentiation of the total energy for coupled system, only the term of the wave equation appears explicitly. We first show that the spectrum of the closed-loop system consists of three branches: one branch is basically along the real axis and accumulates to a finite point; the second branch is also along the real line; and the third branch distributes along two parabola likewise symmetric with the real axis. The asymptotic expressions of both eigenvalues and eigenfunctions are obtained by means of asymptotic analysis. With an estimation of the resolvent operator, the completeness of the root subspace is proved. The Riesz basis property and exponential stability of the system are then concluded. Finally, we show that the associated C0-semigroup is of Gevrey class, which shows that not only the stability but also regularity have been transmitted from regular wave subsystem to the whole system through this boundary connections.

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