One-dimensional Wave Equation Research Articles

Classical Solution of a Problem with Integral Conditions of the Second Kind for the One-Dimensional Wave Equation

for the one-dimensional wave equation given in a half-strip, we consider a boundary value problem with the Cauchy conditions and two nonlocal integral conditions. Each integral condition is a linear combination of a linear Fredholm integral operator along the lateral side of the half-strip applied to the solution and the values of the solution and of a given function on the corresponding base of the half-strip. Under the assumption of appropriate smoothness of the right-hand side of the equation and the initial data, we obtain a necessary and sufficient condition for the existence and uniqueness of a classical solution of this problem and propose a method for finding it in analytical form. A classical solution is understood as a function that is defined everywhere in the closure of the domain where the equation is considered and has all classical derivatives occurring in the equation and in the conditions of the problem.

Inverse problem for one-dimensional wave equation with matrix potential

Abstract We prove a global uniqueness theorem of reconstruction of a matrix-potential a ⁢ ( x , t ) {a(x,t)} of one-dimensional wave equation □ ⁢ u + a ⁢ u = 0 {\square u+au=0} , x > 0 , t > 0 {x>0,t>0} , □ = ∂ t 2 - ∂ x 2 {\square=\partial_{t}^{2}-\partial_{x}^{2}} with zero Cauchy data for t = 0 {t=0} and given Cauchy data for x = 0 {x=0} , u ⁢ ( 0 , t ) = 0 {u(0,t)=0} , u x ⁢ ( 0 , t ) = g ⁢ ( t ) {u_{x}(0,t)=g(t)} . Here u , a , f {u,a,f} , and g are n × n {n\times n} smooth real matrices, det ⁡ ( f ⁢ ( 0 ) ) ≠ 0 {\det(f(0))\neq 0} , and the matrix ∂ t ⁡ a {\partial_{t}a} is known.

Output tracking and disturbance rejection for 1-D anti-stable wave equation

In this paper, we solve the output tracking and disturbance rejection problem for a system described by a one-dimensional anti-stable wave equation, with reference and disturbance signals that belong to W1,∞[0, ∞) and L∞[0, ∞), respectively. Generally, these signals cannot be generated from an exosystem. We explore an approach based on proportional control. It is shown that a proportional gain controller can achieve exponentially the output tracking while rejecting disturbance. Our method consists of three steps: first, we convert the original system without disturbance into two transport equations with an ordinary differential equation by using Riemann variables, then we propose a proportional control law by making use of the properties of transport systems and time delay systems. Second, based on our recent result on disturbance estimator, we apply the estimation/cancellion strategy to cancel to the external disturbance and to track the reference asymptotically. Third, we design a controller using a state observer. Since disturbance does not appear in the observer explicitly (the disturbance is exactly compensated), the controlled output signal is exponentially tracking the reference signal. As a byproduct, we obtain a new output feedback stabilizing control law by which the resulting closed-loop system is exponentially stable using only two displacement output signals.

On the Boundary Controllability of Coupled 1-d Wave Equations

Abstract This paper focuses on the exact controllability of a system of N one-dimensional coupled wave equations where the control is applied on a part of the boundary by a single control. In particular, we consider a coupling matrix A with a single eigenvalue of algebraic multiplicity N. We give a Kalman condition (necessary and sufficient) and give a description of the attainable set. In general, this set is not optimal, but can be refined under certain conditions.

Exact Observability of a 1-Dimensional Wave Equation on a Noncylindrical Domain

We discuss admissibility and exact observability estimates of boundary observation and interior point observation of a one-dimensional wave equation on a time dependent domain for sufficiently regular boundary functions. We also discuss moving observers inside the non-cylindrical domain and simultaneous observability results.

Solving the linear homogeneous one-dimensional wave equation using the Adomian decomposition method

Solving the linear homogeneous one-dimensional wave equation using the Adomian decomposition method

On the 1d wave equation in time-dependent domains and the problem of debond initiation

Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.

Performance Output Tracking for a One-Dimensional Wave Equation With Input Delay

This paper is concerned with the performance output regulation problem for a wave equation with input delay and unmatched disturbance. Firstly, in the case of time delay, the input delay term is translated into a first-order hyperbolic equation and we obtain a cascade system. By applying the method of auxiliary trajectory, the unmatched disturbance is compensated and eliminated. Then, we design a state feedback controller. Meanwhile, with the measured error signal, we construct an observer for the cascade system. Based on the observer, an error feedback controller is developed by replacing the states with their estimations. By using Lyapunov functional method, we also prove the regulation error goes to zero exponentially. Thus, the problem of output tracking is solved for the wave equation in despite of input delay and unmatched disturbance. Finally, the numerical simulations are presented to validate the theoretical results.

Exponential stabilization of solutions for the 1-d transmission wave equation with boundary feedback

The purpose of this work is to study the exponential decay of the energy forthe one-dimensional transmission wave equation with a boundary velocity feedback.Thanks to the perturbed energy method developed by some authors in several contexts, andunder certain conditions, we prove that the feedback controller exponentially stabilizes theequilibrium to zero of the system below, i.e. the feedback leads to faster energy decay.

Optimal Recovery Method of the Solution of a One-Dimensional Wave Equation at Time Τ From Inaccurate Data at t=0 And t = T

The article deals with the problem of optimal recovery of the solution of the wave equation at some time instant by known, but given with some error, functions determining the shape of the string at times t and T. The goal of the paper is to construct an optimal recovery method for the solution of the wave equation from inaccurate data. An important assumption used in the work is the possibility of representing the solution in the form of a Fourier series. The main solution method is the introduction of an auxiliary extremal problem for a conditional extremum, the solution of which determines the optimal recovery method. The result of the work is to find the optimal recovery method among all possible methods. The solution of the restoration problem and the value of the error of optimal recovery are obtained. Cases are indicated when it is possible to reduce the amount of initial information required for solving the problem.

Asymptotics for periodic systems

This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a so-called Ritt operator under a natural ‘resonance’ condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the one-dimensional wave equation with periodic damping.

Parabolic curves in a Helmholtz solution for a bowed string

If one is not familiar with the physics of the violin, it is not easy to guess, even for an experimental physicist, that the so-called Helmholtz motion can be obtained as a solution to the one-dimensional wave equation for the motion of a bowed violin string. It is worth visualising this aspect from a graphical perspective without recourse to ordinary Fourier analysis, as has customarily been done. We show in this paper how to obtain the shape of the Helmholtz trajectory, that is, two mirror-symmetric parabolas, in the ideal case of no losses from internal dissipation and no viscous drag from the air and the non-rigid end supports. We also show that the velocity profile of the Helmholtz motion is also a solution of the one-dimensional wave equation. Finally, we again derive the parabolic shape of the Helmholtz trajectory by applying the principle of energy conservation to a violin string.

Numerical Method for Damping String Vibrations with Unknown Initial State in the Class of Weak Generalized Solutions

The problem of positional boundary control is considered for the spatially one-dimensional wave equation. The objective of control is to transfer the system from an unknown initial state into the state of rest in finite time. A specific feature of the statement of the problem is the weakening of the requirements on the regularity of generalized solutions, observations, and control. The smoothing procedure is used to transfer the problem into the class of strong generalized solutions, where the method previously developed by the authors can be used. The procedure is mathematically justified, and the corresponding results of numerical experiments are given.

Boundary observability of wave equations with nonlinear van der Pol type boundary conditions

In this note we study the boundary observability for one-dimensional wave equation associated with nonlinear boundary condition that can generate complex dynamics. We discuss the exact observability and approximate observability, respectively, in terms of three different types of common boundary observations by studying the wave interactions on the boundary directly.

WKB solutions to the quasi 1-D acoustic wave equation in ducts with non-uniform cross-section and inhomogeneous mean flow properties – Acoustic field and combustion instability

Linear modal analysis based on the spatio-temporal harmonic representation of acoustic fluctuations is a well-known reduced order method to predict combustion instability frequencies. However, a limitation of this method is that it is only applicable to domains with homogeneous mean properties. The Wentzel-Kramers-Brillouin (WKB) method facilitates the development of approximate analytical solutions to the one-dimensional (1-D) acoustic wave equation in domains with inhomogeneous mean properties. The “classical” form of the WKB solution to the wave equation is based on the assumption of high frequency (or, short wavelength), and is applicable to uniform cross-section ducts with a mean temperature gradient, but no mean flow. Subsequently, the WKB method was extended to include uniform mean velocity as well. In the current study, we derive a WKB-type solution to the generalized Helmholtz equation in quasi 1-D ducts with non-uniform cross-sectional areas and inhomogeneities in mean flow properties such as the velocity, temperature, density, and pressure. The WKB solution is applicable under the assumptions of high frequency and slowly varying mean properties. In deriving this solution, a novel approach has been developed to relate the first and second spatial derivatives of the axial velocity fluctuations to the pressure fluctuations (and its derivatives).A second WKB-type solution (referred to as the WKB2 solution) is also obtained by relaxing the slowly-varying mean property assumption, but retaining the high frequency approximation. Thirdly, the Helmholtz equation for pressure is solved numerically without invoking the two WKB assumptions. The WKB, WKB2 and numerical solutions are compared for a number of cases involving combinations of non-uniform duct areas and axially varying mean properties. Finally, we apply the WKB solution to predict the longitudinal combustion instabilities in a 1-D combustor characterized by a sudden expansion, with uniform mean flow in each of the two ducts comprising the combustor, and a linear temperature profile downstream of a compact, planar flame.

On the Solvability of a Mixed Problem for an One-dimensional Semilinear Wave Equation with a Nonlinear Boundary Condition

In this paper, for an one-dimensional semilinear wave equation we study a mixed problem with a nonlinear boundary condition. The questions of uniqueness and existence of global and blow-up solutions of this problem are investigated, depending on the nonlinearity nature appearing both in the equation and in the boundary condition.

Numerical methods for accurate description of ultrashort pulses in optical fibers

We consider a one-dimensional first-order nonlinear wave equation, the so-called forward Maxwell equation (FME), which applies to a few-cycle optical pulse propagating along a preferred direction in a nonlinear medium, e.g., ultrashort pulses in nonlinear fibers. The model is a good approximation to the standard second-order wave equation under assumption of weak nonlinearity and spatial homogeneity in the propagation direction. We compare FME to the commonly accepted generalized nonlinear Schrödinger equation, which quantifies the envelope of a quickly oscillating wave field based on the slowly varying envelope approximation. In our numerical example, we demonstrate that FME, in contrast to the envelope model, reveals new spectral lines when applied to few-cycle pulses. We analyze and compare pseudo-spectral numerical schemes employing symmetric splitting for both models. Finally, we adopt these schemes to a parallel computation and discuss scalability of the parallelization.

On the Cauchy problem for a nonlinear variational wave equation with degenerate initial data

This paper is focused on a one-dimensional nonlinear variational wave equation which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the local existence and uniqueness of classical solutions to its Cauchy problem with initial data given on the parabolic degenerating line.

A weighted finite element mass redistribution method for dynamic contact problems

This paper deals with a one-dimensional wave equation being subjected to a unilateral boundary condition. An approximation of this problem combining the finite element and mass redistribution methods is proposed. The mass redistribution method is based on a redistribution of the body mass such that there is no inertia at the contact node and the mass of the contact node is redistributed on the other nodes. The convergence as well as an error estimate in time is proved. The analytical solution associated with a benchmark problem is introduced and it is compared to approximate solutions for different choices of mass redistribution. However some oscillations for the energy associated with approximate solutions obtained for the second order schemes can be observed after the impact. To overcome this difficulty, a new unconditionally stable and a very lightly dissipative scheme is proposed.

Solution of interval shallow water wave equations using homotopy perturbation method

PurposeThis paper aims to present solutions of uncertain linear and non-linear shallow water wave equations. The uncertainty has been taken as interval and one-dimensional interval shallow water wave equations have been solved by homotopy perturbation method (HPM). In this study, basin depth and initial conditions have been taken as interval and the single parametric concept has been used to handle the interval uncertainty.Design/methodology/approachHPM has been used to solve interval shallow water wave equation with the help of single parametric concept.FindingsPreviously, few authors found solution of shallow water wave equations with crisp basin depth and initial conditions. But, in actual sense, the basin depth, as well as initial conditions, may not be found in crisp form. As such, here these are considered as uncertain in term of intervals. Hence, interval linear and non-linear shallow water wave equations are solved in this study using single parametric concept-based HPM.Originality/valueAs mentioned above, uncertainty is must in the above-titled problems due to the various parametrics involved in the governing differential equations. These uncertain parametric values may be considered as interval. To the best of the authors’ knowledge, no work has been reported on the solution of uncertain shallow water wave equations. But when the interval uncertainty is involved in the above differential equation, then direct methods are not available. Accordingly, single parametric concept-based HPM has been applied in this study to handle the said problems.