Problem For Wave Equation Research Articles

On initial-boundary value problems for one-dimension semilinear wave equations with null conditions

We consider the initial-boundary value problem on R+×R+ for one-dimension systems of semilinear wave equations with null conditions. We show that for homogeneous Dirichlet or Neumann boundary values and sufficiently small initial data, classical solutions always globally exist. Furthermore, we also prove that the global solution is asymptotically free in the energy sense.

Influence of the hidden regularity on the stability of partially damped systems of wave equations

We consider the stability of a system of two wave equations with only one boundary feedback and we show that the stability of the partially damped system depends on the transmission of energy between the two equations. The study confirms that the hidden regularity is an essential ingredient for the stability property. In particular, using a sharp regularity for Neumann problem of wave equation, we improve the usual results on the energy decay rate. This new approach can certainly be applied to other situations of partially damped systems.

On the computation of Nash and Pareto equilibria for some bi-objective control problems for the wave equation

This paper deals with the numerical implementation of a systematic method for solving bi-objective optimal control problems for wave equations. More precisely, we look for Nash and Pareto equilibria which respectively correspond to appropriate noncooperative and cooperative strategies in multi-objective optimal control. The numerical methods described here consist of a combination of the following: finite element techniques for space approximation; finite difference schemes for time discretization; gradient algorithms for the solution of the discrete control problems. The efficiency of the computational methods is illustrated by the results of some numerical experiments.

The null-timelike boundary problems of linear wave equations in asymptotically anti-de Sitter spacetime

In this paper, we study the initial-boundary problem of the massive wave equation which is conformal regular on an asymptotically anti-de Sitter spacetime, where the initial-boundary data are given on an outgoing null hypersurface and a timelike hypersurface, and the asymptotic information is given on the future null infinity. After a conformal rescaling, this problem will reduce to a null-timelike boundary problem of linear wave equation on the conformal compactificated spacetime, where the initial data is given on a null hypersurface and the boundary data are given on two timelike hypersurfaces. We demonstrate well posedness for the associated null-timelike boundary problems. The proofs rely on energy estimates (weighted energy estimates for null-timelike boundary problems) and local existence of solution for initial-boundary problems. This is a toy model motivated by AdS/CFT correspondence.

Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations

Numerical analysis is the area of mathematics that creates, analyzes, and implements algorithms for solving numerically the problems from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. Till now, numerous numerical methods have been introduced. Spectral method is one of those techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform (FFT). This study presents some of the fundamental ideas of spectral method. Orthogonal basis are used to establish computational algorithm. The accuracy and efficiency of proposed model are discussed. This manuscript estimates for the error between the exact and approximated discrete solutions. This paper shows that, grid points for polynomial spectral methods should lie approximately in a minimal energy configuration associated with inverse linear repulsion between points. The wave equation, linear and non-linear boundary value problems are solved using spectral method on the platform of MATLAB language.
 GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 28-42

A novel method for investigation of acoustic and elastic wave phenomena using numerical experiments

The emergence of new types of composite materials, the depletion of existing hydrocarbon deposits, and the increase in the speed of trains require the development of new research methods based on wave scattering. Therefore, it is necessary to determine the laws of wave scattering in inhomogeneous media. We propose a method that combines the advantages of a numerical simulation with an analytical study of the boundary value problem of elastic and acoustic wave equations. In this letter we present the results of the study using the proposed method: the formation of a response from a shear wave in an acoustic medium and the formation of shear waves when a vertically incident longitudinal wave is scattered by a vertical gas-filled fracture. We have obtained a number of analytical expressions characterising the scattering of these wave types.

Control of blowup and approximate optimality conditions for the wave equations

We study the blowup problems for semilinear wave equations with control function in the domain and on the boundary. Sufficient optimality conditions for controlled blowup time are derived in terms of new dynamic programming methodology. We define ε-optimal value function and we construct sufficient ε-optimal conditions for that function again in terms of new dynamic programming inequality.

The lifespan of solutions of semilinear wave equations with the scale-invariant damping in two space dimensions

In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed.The key fact is that, only in two space dimensions, such a special constant in the damping term is a threshold between “wave-like” domain and “heat-like” domain. As a result, we obtain a new type of estimate especially for the critical exponent.

A novel homogenization function method for inverse source problem of nonlinear time-fractional wave equation

A novel numerical technique is developed in this paper to accurately and efficiently resolve the inverse source problem of the nonlinear time-fractional wave equation. Based on all given conditions, the homogenization function of nonlinear time-fractional wave equation can be derived, and then a family of homogenization functions is obtained. Furthermore, a numerical model is established by the superposition of homogenization functions and used for tackling inverse source problem. The proposed method is free of mesh generation, numerical integration, iteration, regularization and fundamental solutions, and it is easy to program and implement on the existing software. Three numerical experiments demonstrate the accuracy and convergence of the proposed strategy for the inverse source problem even with high noise imposed on the boundary conditions.

Magnetic Resonance-Electrical Properties Tomography by Directly Solving Maxwell’s Curl Equations

Magnetic Resonance-Electrical Properties Tomography (MR-EPT) is a method to reconstruct the electrical properties (EPs) of bio-tissues from the measured radiofrequency (RF) field in Magnetic Resonance Imaging (MRI). Current MR-EPT approaches reconstruct the EP profile by solving a second-order partial differential wave equation problem, which is sensitive to noise and can induce large reconstruction artefacts near tissue boundaries and areas with inaccurate field measurements. In this paper, a novel MR-EPT approach is proposed, which is based on a direct solution to Maxwell’s curl equations. The distribution of EPs is calculated by iteratively fitting the RF field calculated by the finite-difference-time-domain (FDTD) technique to the measured values. To solve the time-consuming problem of the iterative fitting, a graphics processing unit (GPU) is used to accelerate the FDTD technique to process the field calculation kernel. The new EPT method was evaluated by investigating a numerical head phantom, and it was found that EP values can be accurately calculated and were relatively insensitive to simulated RF field errors. The feasibility of the proposed method was further validated using phantom-based experimental data collected from a 9.4 Tesla (T) Magnetic Resonance Imaging (MRI) system. The results also indicated that more accurate EP values could be reconstructed from the new method compared with conventional methods. Moreover, even in the absence of phase information of the RF field, the proposed approach is still capable of offering robust EPT solutions, thus having improved practicality for potential clinical applications.

An H2N2 Interpolation for Caputo Derivative with Order in (1, 2) and Its Application to Time-Fractional Wave Equations in More Than One Space Dimension

In this paper, a new derived method is developed for a known numerical differential formula of the Caputo fractional derivative of order $$\gamma \in (1,2)$$ (Li and Zeng in Numerical methods for fractional calculus. Chapman & Hall/CRC numerical analysis and scientific computing, CRC Press, Boca Raton, 2015) by means of the quadratic interpolation polynomials, and a concise expression of the truncation error is given. This new method will be called as the H2N2 method because of the application of the quadratic Hermite and Newton interpolation polynomials. A finite difference scheme with a second order accuracy in space and a $$(3-\gamma )$$-th order accuracy in time based on the H2N2 method is constructed for the initial boundary value problem of time-fractional wave equations. The stability and convergence of the difference scheme are proved. Furthermore, in order to increase computational efficiency, using the sum-of-exponentials to approximate the kernel $$t^{1-\gamma }$$, a fast difference scheme is presented. The problem with weak regularity at the initial time is also discussed with the help of the graded meshes. At each time level, the difference scheme is solved with a fast Poisson solver. Numerical results show the effectiveness of the two difference schemes and confirm our theoretical analysis.

On Stationary Nonequilibrium Measures for Wave Equations

In the paper, the Cauchy problem for wave equations with constant and variable coefficients is considered. We assume that the initial data are a random function with finite mean energy density and study the convergence of distributions of the solutions to a limiting Gaussian measure for large times. We derive the formulas for the limiting energy current density (in mean) and find a new class of stationary nonequilibrium states for the studied model.

Strauss exponent for semilinear wave equations with scattering space dependent damping

It has been asserted that the semilinear wave equation with scattering damping admits the same Strauss critical exponent as the classical semilinear wave equation. In this paper, we devote to showing that this assertion holds for the space dependent damping wave equation when the decay rate of the space dependent variable coefficients of the damping is greater than 2. In particular, we derive the lifespan upperbound estimates for both sub-critical and critical exponents. Moreover, when the nonlinear term is of derivative type, we also obtain the same Glassey exponent and the same upper bound lifespan as that of the corresponding problem for wave equation without damping. Both results show the “hyperbolicity” of the equation from the blow-up aspect.

On the obstacle problem for the 1D wave equation

Our goal is to review the known theory on the one-dimensional obstacle problem for the wave equation, and to discuss some extensions. We introduce the setting established by Schatzman within which existence and uniqueness of solutions can be proved, and we prove that (in some suitable systems of coordinates) the Lipschitz norm is preserved after collision. As a consequence, we deduce that solutions to the obstacle problem (both simple and double) for the wave equation have bounded Lipschitz norm at all times. Finally, we discuss the validity of an explicit formula for the solution that was found by Bamberger and Schatzman.

Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems

<p style="text-indent:20px;">We study the initial boundary value problem of linear homogeneous wave equation with dynamic boundary condition. We aim to prove the finite time blow-up of the solution at critical energy level or high energy level with the nonlinear damping term on boundary in control systems.</p>

Remarks on nonlinear elastic waves with null conditions

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math.142(2000) 225–250) and T. C. Sideris (Ann. Math.151(2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

Elastic and acoustic approximations for solving direct problems of human head ultrasonic study

Abstract In order to obtain an ultrasound image of the human head, it is necessary to solve the corresponding inverse problem. In this case, acoustic approximation is usually considered in order to save computational resources. The purpose of this paper is to figure out which wave phenomena are neglected and whether it is advisable to use the acoustic approximation when studying human head using ultrasound. For this, we performed a comparison of elastic and acoustic wave phenomena using the method Wave Logica of studying solutions of the boundary-value problem of wave equation. This method is based on a combination of classical analytical approaches and analysis of results of computational experiments. To conduct computational experiments, we used the grid-characteristic numerical method on structured curvilinear computational grids in order to take into account the complex multilayer structure of the human head. Also accurate transformation of elastic and acoustic wave equations from three-dimensional case to two-dimensional case and accurate formula for pressure in elastic media in two-dimensional case are discussed in the paper.

Uniform Energy Decay for Wave Equations with Unbounded Damping Coefficients

We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence result of a weak solution. In this case we never impose strong assumptions such as compactness of the support of the initial data. This means that we never rely on the finite propagation speed property of the solution, and we try to deal with an essential unbounded coefficient case.

On a backward problem for two-dimensional time fractional wave equation with discrete random data

This paper is concerned with a backward problem for a two- dimensional time fractional wave equation with discrete noise. In general, this problem is ill-posed, therefore the trigonometric method in nonparametric regression associated with Fourier truncation method is proposed to solve the problem. We also give some error estimates and convergence rates between the regularized solution and the sought solution under some assumptions.