Wave Equation Research Articles

Numerical integrator for highly oscillatory differential equations based on the Neumann series

AbstractWe propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

The Blow-Up of the Local Energy Solution to the Wave Equation with a Nontrivial Boundary Condition

In this study, we examine the wave equation with a nontrivial boundary condition. The main target of this study is to prove the local-in-time existence and the blow-up in finite time of the energy solution. Through the construction of an auxiliary function and the imposition of appropriate conditions on the initial data, we establish the both lower and upper bounds for the blow-up time of the solution. Meanwhile, based on these estimates, we obtain the result of the local-in-time existence and the blow-up of the energy solution. This approach enhances our understanding of the dynamics leading to blow-up in the considered condition.

A Novel Method for Predicting the Behavior of a Sucker Rod Pumping Unit Based on the Polished Rod Velocity

Fault dynamometer cards are the basis of the diagnosis technique for sucker rod pumping systems. Predicting fault cards with a pumping condition model is an economical and effective method. The usual model is described by a mixed function of the pump displacement and pump load, and it is difficult to use in the prediction method based on the analytical solution of the sucker rod string wave equation. In this paper, a normal pumping condition model described by a function of polished rod velocity is proposed. For the analytical solution of the sucker rod wave equation, an iterative prediction algorithm with pumping condition models is proposed, its convergence is analyzed, and then it is validated by classical finite difference method simulated cards and measured surface dynamometer cards. The results show that the proposed algorithm is accurate. The algorithm has a maximum relative error of 0.10% for the classical method simulated card area and 1.45% for the measured card area. The research of this paper provides an effective scheme for the design, prediction, and fault diagnosis of a sucker rod pumping system with an analytical solution.

Magnetic flux-driven modulation of Weyl pair dynamics on catenoid bridge: A theoretical analysis

In this study, we examine the behavior of a Weyl pair influenced by magnetic flux on a catenoid bridge by obtaining analytical solutions of the corresponding fully covariant two-body Dirac equation. By deriving a non-perturbative wave equation governing their relative motion, we obtain a solution expressed in terms of the Confluent Heun function. Our findings indicate that the dynamics of this pair are influenced not only by the magnetic flux but also by the radius (ρ) of the catenoid bridge. We observe that the ground state energy of the Weyl pair can vary significantly, shifting from 0.258eV to 1.8 meV as the ρ value changes from 10nm to 1 μm. Furthermore, our results show that the Weyl pair’s behavior can resemble that of either a single fermion or boson, and intriguingly, this behavior can be modulated by adjusting the magnetic flux.

On one solution of the problem of vibrations of mechanical systems with moving boundaries

An analytical method of solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that makes it possible to approximate quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. An estimate of the errors of the approximatemethod was made depending on the speed of the boundary movement.

Vortex position detection using a scanning triangular aperture in a digital micromirror device

Optical vortices, as solutions to the wave equation, remain stable under ideal theoretical conditions. However, in experiments where beams produced with holograms and spatial light modulators introduce perturbations, their phase structures can be disrupted, leading to instability. Specifically, higher-order optical vortices (with topological charge |ℓ|>1 ) are inherently unstable in that they tend to separate in a series of optical vortices with unit topological charge (|ℓ|=1) even with small perturbations. In this work, we demonstrate a technique to detect the positions of optical vortices using a scanning triangular aperture in a digital micromirror device and a digital pinhole. We observe the diffraction patterns of a vortex through the triangular aperture, and we show that we can determine the center of an optical vortex (OV) by measuring the intensity at the center of a pattern using a digital pinhole. We investigate the changes in the measured signal for different triangular apertures and pinhole sizes. We observe that while smaller pinholes lead to a decrease in light intensity, the detected position of a single OV with unit topological charge is similar for all pinhole sizes. We also find that the triangular aperture size becomes important when locating vortex pairs. Using a triangular aperture with a radius R = 0.52 mm, we can resolve OV pairs at least 86.4μ m apart in our experiments. Our methods help pave the way to understanding the fundamental behavior of multiple vortex interactions in optical beams and also can be used when determining the position of an OV in metrology.

A preserving accuracy two-grid reduced-dimensional Crank-Nicolson mixed finite element method for nonlinear wave equation

In this paper, we mainly resort to a proper orthogonal decomposition (POD) method to study the reduced-dimension of unknown mixed finite element (MFE) solution coefficient vectors in the two-grid Crank-Nicolson MFE (TGCNMFE) method for the nonlinear wave equation and build a preserving accuracy reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method for the nonlinear wave equation. For this purpose, we first build a new TGCNMFE method for the nonlinear wave equation and demonstrate the existence, unconditional stability, and error estimates for the TGCNMFE solutions. From there, we build a preserving accuracy RDETGCNMFE method for the nonlinear wave equation by using the POD method to decrease the dimension of unknown TGCNMFE solution coefficient vectors and employ the matrix analysis to analyze the existence, unconditional stability, convergence, and errors for the RDETGCNMFE solutions. We finally make use of numerical experiments to verify our theoretical results and to show the advantages of RDETGCNMFE method.

Measurement and modeling of strain waves in germanium induced by ultrafast laser pulses

Transient reflectivity measurements are used to probe the strain waves induced by ultrashort laser pulses in bulk [100] germanium. The measurement signals are compared to purely analytical model functions based on the known material parameters for germanium. The modeling includes (i) a derivation of analytical solutions to the wave equation for strain waves coupled to the diffusion equation for heat and charge carriers and (ii) an expression for the impact on reflection coefficients that are caused by perturbations to the dielectric function but extended to cover a non-isotropic, uniaxial dielectric tensorial form. The model is held up against transient reflectivity measurements with an s- and a p-polarized probe and with a probe wavelength in the range of 502–710 nm. Excellent agreement is found when comparing the oscillatory shape of the measurement signals to the models. As for the magnitude of the oscillations, the models reproduce the overall trends of the experiment when using the previously published values for the elasto-optical tensor measured under static strain.

Polynomial Energy Decay Rate for the Wave Equation with Kinetic Boundary Condition

Polynomial Energy Decay Rate for the Wave Equation with Kinetic Boundary Condition

Exponential convergence to steady-states for trajectories of a damped dynamical system modeling adhesive strings

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.

Frozen Gaussian sampling for scalar wave equations

Frozen Gaussian sampling for scalar wave equations

Viscoacoustic VTI media reverse time migration method in the complex-domain based on the staining algorithm

Underground media exhibit characteristics of viscosity and anisotropy, with vertical transversely isotropic (VTI) media being a common example of anisotropic media. Consequently, it is crucial to address the viscoacoustic VTI wave equation during forward modeling and migration imaging processes to attain highly precise imaging outcomes. However, conventional subsalt imaging techniques yield low accuracy and suffer from a low signal-to-noise ratio. To overcome this challenge, this study proposes a staining algorithm-based viscoacoustic vertical transversely isotropic reverse time migration (VTI RTM) in the complex-domain method by deriving the complex-domain viscoacoustic VTI equation. This method requires both the traditional real velocity field and the imaginary velocity field as inputs. The imaginary velocity field contains stained geological bodies and significant areas without data. Seismic waves propagate normally in the real wavefield, while in the imaginary wavefield, reflections occur only when the wavefront encounters stained geological bodies. The proposed method is assessed through testing on both a horizontally layered model and a salt model. Through comparative analysis, it has been verified that the method fully incorporates reflected waves with distinct structural features into the imaginary wavefield. Numerical experiments demonstrate that this method can accurately image subsalt structures.

Polynomial stabilization of the wave equation with a time varying delay term in the dynamical control

We consider the one-dimensional wave equation with a time-varying delay term in the dynamical control. Under suitable assumptions, we show the well posedness of the problem. These results are obtained by using semi-group theory. Combining the multiplier method with a non linear integral inequality, a rational energy decay result of the system is established. A fundamental aspect of this paper is that our involved operator is time dependent, therefore the standard frequential method cannot be invoked.

Dynamics of the traveling wave solutions of conformable time-fractional ISLW and DJKM equations via a new expansion method

This work searches the new wave solutions of the time-fractional ion sound and Langmuir waves (ISLWs) equation and the time-fractional Date–Jimbo–Kashiwara–Miwa (DJKM) equation with the conformable derivative. The m+1G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( m+\\frac{1}{G^\\prime }\\right)$$\\end{document}-expansion method is employed for these models for the first time. Getting support from symbolic software, some new solutions are presented, which will be constructive for studies in the field of nonlinear physical phenomena. New wave solutions obtained by using the m+1G′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( m+\\frac{1}{G^\\prime }\\right)$$\\end{document}-expansion method have applications in engineering, physics, and other scientific fields. The regarded method may provide a framework for the exact solutions of several evolution equations in applied sciences. Moreover, the graphical illustrations are also demonstrated to observe the dynamical behaviours of the solutions, which may benefit the development of new technologies.

Dispersion of waves propagating in the ice-covered Arctic Ocean

The Arctic Ocean was modeled as an ice–seawater–sediment system, where the ice cover and seawater were assumed to be inhomogeneous solid and liquid, respectively, while the sediment was assumed to be homogeneous liquid. Transfer matrixes relating the displacements and stresses at the lower surface and those at the upper surface for a thin solid layer, and a thin liquid layer were derived. Furthermore, a dispersion equation for waves propagating in the ice-covered Arctic Ocean was derived using the transfer matrix technique. The phase- and group-velocity dispersion curves were obtained by solving the dispersion equation numerically. The results show that the dispersion curves for the Arctic Ocean with ice cover are much more complex than those without ice cover. Except for the new mode, the phase-velocity curve for the n-th (n > 2) mode exhibited a slight distortion, which caused a sharp peak in the group-velocity curve. These peak values, which depend on the order of the mode, may be significantly higher than the speed of sound in seawater. The variation of the ice cover thickness had significant influence on the dispersion curves of the first and second modes. Moreover, the influence of the seawater depth on the dispersion curves were investigated.

Thermodynamics of harmonic oscillator with minimal length

The generalized uncertainty principle has been applied to the Schrödinger wave equation for a one-dimensional harmonic oscillator to generate a sixth-order generalized Schrödinger equation in the position representation. The energy eigenvalues and the eigenfunctions of the sixth-order equation have been obtained. The results show the approximate correction terms of the energies due to the modified uncertainty principle. The quantum partition functions derived from the energy eigenvalue have also been used to study the thermodynamic properties of the system. The results suggest a lower bound for the minimal length equivalent to the thermal wavelength of the oscillator at very high temperature.

Norm inflation for the viscous nonlinear wave equation

Norm inflation for the viscous nonlinear wave equation

The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

An exploration of the (3+1)-dimensional negative order KdV-CBS model: Wave solutions, Bäcklund transformation, and complexiton dynamics.

This research paper focuses on the study of the (3+1)-dimensional negative order KdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, an important nonlinear partial differential equation in oceanography. The primary objective is to explore various solution techniques and analyze their graphical representations. Initially, two wave, three wave, and multi-wave solutions of the negative order KdV CBS equation are derived using its bilinear form. This analysis shed light on the behavior and characteristics of the equation's wave solutions. Furthermore, a bilinear Bäcklund transform is employed by utilizing the Hirota bilinear form. This transformation yields exponential and rational function solutions, contributing to a more comprehensive understanding of the equation. The resulting solutions are accompanied by graphical representations, providing visual insights into their structures. Moreover, the extended transformed rational function method is applied to obtain complexiton solutions. This approach, executed through the bilinear form, facilitated the discovery of additional solutions with intriguing properties. The graphical representations, spanning 2D, 3D, and contour plots, serve as valuable visual aids for understanding the complex dynamics and behaviors exhibited by the equation's solutions.

Real and damped modes for an interacting fermion–antifermion pair: exciton in monolayer medium

In this manuscript, we investigate the interaction between a fermion–antifermion pair in (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensions, governed by a Coulomb-type inter-particle potential. We explore analytical solutions for a fully covariant two-body Dirac equation derived from quantum electrodynamics, focusing on a spinless composite system. We present a non-perturbative second-order wave equation and derive a solution using the generators of the sℓ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\ell _{2}$$\\end{document} algebra. Subsequently, we determine the relativistic frequency modes of the system and extend our findings to electron–hole pairs in a monolayer medium in the vicinity of substrate. We analyze the impact of the effective dielectric constant (ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon$$\\end{document}) of the surrounding medium on the real and damped modes of an exciton. Our results indicate that the ground state binding energy of the system can vary from the order of eV to a few meV, depending on changes in ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon$$\\end{document}. Additionally, we observe that the decay time for the ground state of such an exciton is on the order of ∼10-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim 10^{-12}$$\\end{document} s. These findings suggest that the evolution of such a pair can be controlled by adjusting the effective dielectric constant of the surrounding medium.