Fourth-order Wave Research Articles

A class of fourth-order dispersive wave equations with exponential source

A class of fourth-order dispersive wave equations with exponential source

Unconditional analysis of the linearized second-order time-stepping scheme combined with a mixed element method for a nonlinear time fractional fourth-order wave equation

In this article, we present a numerical algorithm for solving a two-dimensional nonlinear fourth-order time fractional wave model, where a Galerkin mixed finite element method yielded by introducing two auxiliary variables v=ut and σ=Δu−f(u) is used in the spatial direction and a second-order θ scheme with the weighted shifted Grünwald difference (WSGD) formula is applied in the time direction. Utilizing the space-time splitting technique without limiting the relationship between spatial mesh size and temporal step size, we derive the unconditional optimal error estimate in L2-norm and the stability result. Further, for verifying the feasibility and effectiveness of our algorithm with or without starting parts, we implement numerical calculations by choosing nonsmooth solutions.

Potential well method for a class of fourth order wave equations with Newtonian potential

Potential well method for a class of fourth order wave equations with Newtonian potential

Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity

Lifespan estimates and asymptotic stability for a class of fourth-order damped p-Laplacian wave equations with logarithmic nonlinearity

Numerical solution of the cavity scattering problem for flexural waves on thin plates: Linear finite element methods

Flexural wave scattering plays a crucial role in optimizing and designing structures for various engineering applications. Mathematically, the flexural wave scattering problem on an infinite thin plate is described by a fourth-order plate wave equation on an unbounded domain, making it challenging to solve directly using the regular linear finite element method (FEM). In this paper, we propose two numerical methods, the interior penalty FEM (IP-FEM) and the boundary penalty FEM (BP-FEM) with a transparent boundary condition (TBC), to study flexural wave scattering by an arbitrary-shaped cavity on an infinite thin plate. Both methods decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary. A TBC is then constructed based on the analytical solutions of the Helmholtz and modified Helmholtz equations in the exterior domain, effectively truncating the unbounded domain into a bounded one. Using linear triangular elements, the IP-FEM and BP-FEM successfully suppress the oscillation of the bending moment of the solution on the cavity boundary, demonstrating superior stability and accuracy compared to the regular linear FEM when applied to this problem.

A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations

Abstract This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.

Efficient high-order physical property-preserving difference methods for nonlinear fourth-order wave equation with damping

Efficient high-order physical property-preserving difference methods for nonlinear fourth-order wave equation with damping

The Galerkin method and hinged beam dynamics

In this paper we prove large-time existence and uniqueness of high regularity weak solutions to some initial/boundary value problems involving a nonlinear fourth order wave equation. These sorts of problems arise naturally in the study of vibrations in beams that are hinged at both ends. The method used to prove large-time existence is the Galerkin approximation method.

Higher-order energy-preserving difference scheme for the fourth-order nonlinear strain wave equation

This paper focus on construction of high-order energy-preserving difference scheme for the fourth-order nonlinear strain wave equation with an energy conservation law. This target model is firstly transformed into an equivalent system by using the method of trigonometric scalar auxiliary variables. The resulting equivalent system possess a modified energy conservation law, and a fourth-order difference scheme with analogously discrete energy conservation law is developed based on the resulting equivalent system. The boundedness and convergence of the numerical solutions in the maximum norm are shown. The effectiveness of the difference scheme is verified by several numerical experiments.

Long-time behavior for fourth order nonlinear wave equations with dissipative and dispersive terms

The initial boundary value problem for a class of fourth-order nonlinear wave equations with dissipative terms and dispersive terms is considered in this paper. We first establish the global existence and the exponential decay of the solution. Next, based on Condition (C) and semigroup theory arguments, the existence of the global attractor is derived.

Blowup phenomena for a fourth order wave equation at high initial energy level

Blowup phenomena for a fourth order wave equation at high initial energy level

Improved blowup time estimates for fourth-order damped wave equations with strain term and arbitrary positive initial energy

We propose a new differential inequality that improve the upper bound of the blowup time estimate for nonlinear fourth-order damped wave equations with strain term and arbitrary positive initial energy. We also give two new initial conditions to expand the range of the initial data leading to the finite time blowup of solutions. We obtain a sharp result of finite time blowup for the special case of the new differential inequality.We illustrate our results with some simulations.

Global existence and blow-up of solutions for coupled bi-harmonic nonlinear wave equations

Abstract This work studies the coupled nonlinear fourth-order wave system u ¨ i + Δ 2 ⁢ u i + u i = ± ( ∑ 1 ≤ j ≤ m a i ⁢ j ⁢ | u j | p ) ⁢ | u i | p - 2 ⁢ u i . \ddot{u}_{i}+\Delta^{2}u_{i}+u_{i}=\pm\bigg{(}\sum_{1\leq j\leq m}a_{ij}\lvert u% _{j}\rvert^{p}\biggr{)}\lvert u_{i}\rvert^{p-2}u_{i}. The main goal is to develop a local theory in the energy space and to investigate some issues of the global theory. Indeed, using a standard contraction argument coupled with Strichartz estimates, one obtains a local solution in the inhomogeneous Sobolev space ( H 1 ) m {(H^{1})^{m}} for the energy sub-critical regime. Then the local solution extends to a global one in the attractive regime; also in the energy critical case there is a global solution with small data. For a repulsive source term, by using the potential well theory with a concavity argument, the local solution may concentrate in finite time or extend to a global one. Finally, in the inter-critical regime, one proves the existence of infinitely many non-global solutions with data near to the stationary solution. Here, one needs to deal with the coupled source term which gives some technical restrictions such as p ≥ 2 {p\geq 2} in order to avoid a singularity. This assumption in the inter-critical regime gives a restriction on the space dimension.

An energy-conserving finite element method for nonlinear fourth-order wave equations

In this paper, an energy-conserving finite element method is developed and intensively analyzed for a class of nonlinear fourth-order wave equations in a general sense for the first time, where the two-level, Crank-Nicolson type of temporal discretization scheme is designed to cooperate with the Lagrange finite element approximation in space in order to achieve the conservation of discrete energy at each time step. The energy conservation is crucial in engineering field to preserve the total energy as constant for dynamic vibration problems of beams and thin plates that can be modeled by the presented wave equations. In addition, the optimal spatial convergence properties in both L2- and H1-norm and the second-order temporal approximation rate are obtained for finite element solutions of u (deflection), Δu (bending moment) and/or ut (deflection speed) at the same time. Numerical experiments are carried out to validate all attained theoretical results.

Electromagnetic wave propagation in general Kasner-like metrics

The curved spacetime Maxwell equations are applied to the anisotropically expanding Kasner metrics. Using the application of vector identities we derive second-order differential wave equations for the electromagnetic field components; through this explicit derivation, we find that the second-order wave equations are not uncoupled for the various components (as previously assumed), but that gravitationally induced coupling between the electric and magnetic field components is generated directly by the anisotropy of the expansion. The lack of such coupling terms in the wave equations from several prior studies may indicate a generally incomplete understanding of the evolution of electromagnetic energy in anisotropic cosmologies. Uncoupling the field components requires the derivation of a fourth-order wave equation, which we obtain for Kasner-like metrics with generalized expansion/contraction rate indices. For the axisymmetric Kasner case, [Formula: see text], we obtain exact field solutions (for general propagation wave vectors), half of which appear not to have been found before in previous studies. For the other axisymmetric Kasner case, [Formula: see text], we use numerical methods to demonstrate the explicit violation of the geometric optics approximation at early times, showing the physical phase velocity of the wave to be inhibited towards the initial singularity, with [Formula: see text] as [Formula: see text].

The Fourth-Order Nonlinear Schrödinger Equation and Stability Analysis for Stokes Waves on Slowly Varying Topography

The surface gravity wave equation is expanded to the fourth-order wave steepness on slowly varying topography, obtaining a topographic modified nonlinear Schrödinger (TMNLS) equation. When the time scale is longer than ε-3 times of the dominant wave period or the space scale is larger than ε-3 times the dominant wavelength, the second water depth derivative and the square of the first water depth derivative affect the first-order wave amplitude. The instability area for a uniform Stokes wave train by small perturbations is the entire wavenumber space, except for a specific stability curve on infinite and slowly varying depth. The depth variation terms affect the growth rate of uniform Stokes wave train on the order of 0.01. The stability curve shows more sensitive to the depth variation in x direction than that in y direction. The increment of the value for depth variation in x direction contributes the stable wave number of perturbation to approach or parallel to y axis. The increment of the value for depth variation in y direction helps the stable wave number of perturbation to approach or parallel to x axis.

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation

Finite Time Blowup in a Fourth-Order Dispersive Wave Equation with Nonlinear Damping and a Non-Local Source

In this work, we consider a class of initial boundary value problems for fourth-order dispersive wave equations with superlinear damping and non-local source terms as well as time-dependent coefficients in Ω×(t>0), where Ω is a bounded domain in RN and N≥2. We prove that there exists a safe time interval of existence in the solution [0,T], with T being a lower bound of the blowup time t*. Moreover, we find an explicit lower bound of t*, assuming the coefficients are positive constants.

Efficient numerical algorithm with the second-order time accuracy for a two-dimensional nonlinear fourth-order fractional wave equation

In this article, we construct an efficient numerical algorithm with the second-order time accuracy for a two-dimensional nonlinear fourth-order fractional wave equation. We introduce an auxiliary variable to transform the fractional fourth-order wave problem into a low order coupled system, and then integrate the resulting equations by using Riemann–Liouville fractional integral. For the obtained fractional integral system, we use the Crank–Nicolson scheme in time with the weighted and shifted Grünwald integral (WSGI) approximation and the finite element method in space to arrive at the fully discrete coupled system. We show the detailed numerical algorithms and carry out numerical tests by taking two two-dimensional examples on rectangular and circular space domains to verify the effectiveness and convergence efficiency for our algorithm.

Primary aberration optimization for double-plane symmetric beam shaping systems using a pair of curved reference surfaces.

The geometric aberration of centered refracting double-plane symmetric optical systems (DPSOS) is investigated. For DPSOS with different defocus values in the tangential plane and the sagittal plane (astigmatic wavefront), a pair of curved reference surfaces which vanishes the quadratic terms of the optical path difference (OPD) between a general ray and a reference ray are deduced. With the curved reference surfaces, the primary (fourth-order) wave aberration function for DPSOS is calculated and analyzed, which can be used for beam shaping designs with astigmatic input wavefront, such as slab lasers and semiconductor lasers. Further, the proposed curved reference surfaces can be applied to analyze the aberrations of general DPSOS.