One-dimensional Wave Equation Research Articles

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.

Active disturbance rejection control approach to the output tracking of PDEs with general disturbances: A new perspective

In recent years, significant progress has been made in output regulation for systems described by partial differential equations (PDEs) through the active disturbance rejection control approach (ADRC). However, in all these studies, the disturbance typically manifests in a single channel and requires estimation, often necessitating additional measured signals. This is because the disturbance must be identified through the output. In this paper, we approach this problem from a novel perspective: disturbances occur in all channels, yet few measured signals are required. In most cases, only the tracking error is utilized as the measured signal. This marks a significant advancement in the application of ADRC to PDEs. We primarily demonstrate this approach by addressing the output tracking problem for a multi-dimensional heat equation, considering general disturbances across all possible channels. Additionally, we briefly discuss some other one-dimensional problems studied in the literature, such as the one-dimensional wave equation.

Study of Support Motion of a Finite Bar with a Boundary Damper and a Spring Using Analytical Approaches and FEM

In this paper, we solve the vibration problem of a finite bar with a viscously damped boundary in conjunction with a spring on the same side subject to the support motion on the other side. To avoid the computation of complex modes and the difficulty of orthogonal conditions for the partial differential equation (PDE) of a continuous system, two alternatives are employed. One is the analytical derivation by using the diamond rule of the method of characteristics. The advantage is that this method can yield an analytical solution. However, the disadvantage is that the method can be only applied to some simple problems such as a linear system and the one-dimensional (1D) wave equation. The other numerical method, finite element method (FEM), is incorporated into a general-purpose program to solve this problem. The advantage of the FEM is that this methodology can be applied to solve various problems such as different PDEs. Therefore, the solution obtained by using the FEM is compared with that of the analytical solution. Two special cases, only a spring and a damper alone, are also considered by using the FEM. Interestingly, the same silent area is captured in the displacement profile by using the FEM for the three cases. We also find that the displacement profiles have slope discontinuity which only occurs on the characteristic line. After the wave front arrives at the boundary, various responses appear due to different boundaries, spring, damper and both. It is found that the displacement amplitude of the general case is smaller than the other two special cases. This result matches the application of engineering practice.

Boundary controllability for variable coefficients one-dimensional wave equation with interior degeneracy

Abstract In this paper, we study boundary controllability for the linear extension problem of a wave equation with space-dependent coefficients and having an internal degeneracy. For this purpose, we mainly focus on the well-posedness and the boundary null controllability of a relaxed version of the original problem, namely, to some degenerate transmission problem. The key ingredient is to derive direct and inverse inequalities for the associated homogeneous degenerate adjoint problem. By these inequalities, we deduce that the transmission problem has a unique solution by transposition and this solution is null controllable. Moreover, we give an explicit formula of the controllability time.

Multiple solutions and conserved vectors of a shallow water wave equation arising in fluid mechanics; Lie group analysis

Investigation has shown that shallow water wave model equations constitute a class of hyperbolic partial differential equations describing the movement below a pressure surface in a fluid, and this makes them highly useful to be studied. Therefore, in this paper we analytically investigate a one-dimensional shallow water wave equation that is highly applicable in fluid dynamics. We leverage the robust nature of the Lie group technique to provide symmetries to the equation. Commutator and adjoint representations of symmetries were generated and presented in tabular form. Sequel to that, we perform reductions of the shallow water wave model under study via its point symmetries so that its exact solutions can be found. Thus, to achieve this, various combinations of the symmetries are made. Besides, we examine the travelling wave solutions of the model. The direct integration method is used to obtain elliptic integral functions as well as elliptic solutions in terms of the Weierstrass function. In addition, we utilize standard techniques such as (G′/G)-expansion, power series, and Kudryashov’s approaches to generate more closed-form solutions to the model. Other interesting solutions achieved include hyperbolic, trigonometric, rational, and exponential functions. In search of the physical meaning associated with the results, graphical demonstrations are included, and these are given in 2D, 3D, and density plots. Thereafter, the construction of conserved vectors for the aforementioned model is done with the use of the standard multiplier approach and Ibragimov’s theorem. These techniques give the opportunity to furnish various forms of conserved currents.

Exact Null Controllability of String Equations with Neumann Boundaries

This article focuses on the exact null controllability of a one-dimensional wave equation in noncylindrical domains. Both the fixed endpoint and the moving endpoint are Neumann-type boundary conditions. The control is put on the moving endpoint. When the speed of the moving endpoint is less than the characteristic speed, we can obtain the exact null controllability of this equation by using the Hilbert uniqueness method. In addition, we get a sharper estimate on controllability time that depends on the speed of the moving endpoint.

Global existence and asymptotic profile for a damped wave equation with variable-coefficient diffusion

We considered a Cauchy problem of a one-dimensional semilinear wave equation with variable-coefficient diffusion, time-dependent damping, and perturbations. The global well-posedness and the asymptotic profile are given by employing scaling variables and the energy method. The lower bound estimate of the lifespan to the solution is obtained as a byproduct.
 For more information see https://ejde.math.txstate.edu/Volumes/2024/04/abstr.html

Substructure elimination and binding method for vibration systems governed by a one-dimensional wave equation

Substructure elimination and binding method for vibration systems governed by a one-dimensional wave equation

Specific Point in Time Excitation Control Method for Spatial Multi-Degree-of-Freedom Systems under Continuous Operation

The port container gantry crane studied in this paper is a four-degree-of-freedom spatial continuous system. In actual work, in order to make the container transfer smoothly, the response of the whole system needs to be accurately predicted and timely adjusted. The whole system is divided into rotary mechanism, lifting mechanism, lifting trolley mechanism, and big cart mechanism for detailed analysis. By constructing the field transfer matrix, a one-dimensional wave equation of continuous system and the Lagrange equation with redundant parameters, the response of each subsystem is solved precisely. The results of the study found that in some periods, the swing of the container was too large. In order to improve the safety and stability of transmission, an active control method of specific point in time excitation (SPE) is proposed for the first time. This method predicts the swing amplitude of the container in advance using the response results of the numerical model. When the set response interval is exceeded, the external excitation intervention can effectively inhibit the moving range of the container in the transit process. Finally, the results are compared with the simulation model to achieve the experimental purpose. It is in line with the expected experimental effect.

Accelerating inverse inference of ensemble Kalman filter via reduced-order model trained using adaptive sparse observations

The implementation of data assimilation often struggles with low computational efficiency due to the complexity of high-fidelity numerical modeling and the large size of the observation vectors. To address this challenge, it is necessary to employ reduced-order strategies for both the full-order dynamical model and observation operator. Proper orthogonal decomposition (POD) based reduced-order model (ROM) and discrete empirical interpolation method (DEIM) based sparse observation identification are often considered as primary solutions. However, their effective combination requires the determination of high-quality POD reduced-order basis vectors. The current research focuses on incorporating the ROM and DEIM into a Kalman filter framework, resulting in a reduced-order Kalman filter with sparse observations. The key aspect is identifying the optimal POD basis, which can be achieved iteratively by introducing a relative root-mean-square-error (RMSE) for a near-optimal exploration of both the desired low-order POD basis and observation locations. The singular value decomposition (SVD) of the snapshots is performed in a low-dimensional subspace, leading to significant computational savings. This improved POD basis not only enhances the reconstruction of the ROM, but also helps build the DEIM observation matrix, resulting in simultaneous dimensionality reduction of the state and observation space. The effectiveness of this algorithm is demonstrated through the retrieval of initial conditions for one-dimensional tsunami wave equations with real data scenarios. Experiments are conducted for both smooth and nonsmoothed initial conditions. The results show that when the observation data is available at 3 and 6 spatial observation locations, our method only requires fewer vectors with the first 45 and 90 dominant modes derived using the RMSE, compared to as many as 210 and 252 derived from the relative energy criterion. Our method enables a low-modal ROM to effectively reconstruct the wave height field, leading to considerable prediction capability for potential long-wave dynamics with comparable accuracy. The CPU time savings achieved through our method is at least one to two orders of magnitude compared to the CPU time required by the full-order model.

A mathematical study about the sustaining phenomenon of overtone in Flageolet harmonics on bowed string instruments

Flageolet harmonics is a playing technique, in which a player lightly touches a nodal point on a string with their finger. Previous studies have reported that the harmonic sound sustains for a short time after the finger is removed from the string while flageolet harmonics is performed on bowed string instruments. However, the mechanism of this harmonics-sustaining phenomenon and the parameter dependency of its sustaining time remains unclear. The purpose of this study is to mathematically investigate the dependence of sustaining time on parameters related to violin playing and string characteristics, and thereby, to elucidate the mechanism of the harmonics-sustaining phenomenon. To this end, a mathematical model was devised by incorporating the effects of bowing and touching into a one-dimensional wave equation. Subsequently, numerical simulation were performed to analyze the behavior of the model. The devised model successfully reproduced the harmonics-sustaining phenomenon in which the parameter dependence of sustaining time was qualitatively consistent with the author’s empirical observations. It was found that the parameter dependence of sustaining time follows the power law. Furthermore, dimensional analysis was performed, yielding a formula that expresses the relationship between the sustaining time and the maximum and minimum bow force required to generate Helmholtz motion.

Design and development of a piezoelectric-hydraulic hybrid actuator with quarter-wavelength tubes

The piezoelectric hydraulic actuator is a hybrid device consisting of a hydraulic pump driven by a piezoelectric stack connected to a hydraulic cylinder. For this type of piezoelectric actuator, the inertial force caused by the flow pulsation of the liquid will inhibit the movement of the piezoelectric vibrator and reduce its output performance. To solve these issues, two tubes were embedded into the piezoelectric-hydraulic hybrid actuation system for the first time to act as mechanical bandstop filters by interfering with the propagation of acoustic waves. The acoustic power transmission loss of the tube is derived from the one-dimensional wave equation. According to the experimental results, when the excitation frequency is close to the optimal operating frequency corresponding to the tubes, the liquid pulsation rate is reduced, the influence of inertial force on the actuator is weakened, and the output performance is relatively significantly improved. This strategy finally leads to a maximum no-load velocity of 153.5 mm s−1 and a maximum blocking force of 261.5 N for the hybrid actuator with 300 mm tubes; the maximum no-load velocity and blocking force of the hybrid actuator with 500 mm tubes are 94.45 mm s−1 and 230 N, respectively. Furthermore, this strategy can be used in other electrohydraulic actuators to enhance their capabilities.

Exact Null Controllability of a One-Dimensional Wave Equation with a Mixed Boundary

In this paper, exact null controllability of one-dimensional wave equations in non-cylindrical domains was discussed. It is different from past papers, as we consider boundary conditions for more complex cases. The wave equations have a mixed Dirichlet–Neumann boundary condition. The control is put on the fixed endpoint with a Neumann boundary condition. By using the Hilbert Uniqueness Method, exact null controllability can be obtained.

Time-reversal technique for pipeline defect detection.

In this paper, time-reversal (TR) technique is proposed for the detection of discrete defects (e.g., leaks and blockages) in pressurized fluid-filled pipelines using active transient waves. The proposed technique is based on the one-dimensional wave equation and decouples the search for the location from that of the size of the defect. Another key feature of this method is that the measured pressure head signal can be truncated so as to eliminate undesirable reflections from unknown or uncertain parts of the pipe system so as not to compromise the accuracy and robustness of the detection. The proposed method is tested through numerical simulations for a single leak, multiple leaks, and multiple mixed-type of defects (i.e., leaks and blockages) in elastic as well as visco-elastic pipe systems. The proposed method is also validated through laboratory experiments wherein leaks and blockages of various sizes are considered. The sensitivity analysis of the method with respect to wave speed and signal-to-noise ratio (SNR) is also performed. In all cases, the method has shown accurate results in terms of localization, classification, and size estimation of the defects even for SNR as low as 0 dB.

A preliminary development of a coupled surface and subsurface flow model for swale system

This study presents the preliminary stage of the development of a conjunctive surface-subsurface model for simulating the flow in a swale. The surface flow is modelled by one-dimensional dynamic wave equation. To improve accuracy, a third-order numerical scheme is used to solve the advection terms in the dynamic wave model. Meanwhile, the subsurface flow is modelled as one-dimensional vertical flow through nondeformable porous media without air compression effect. At preliminary stage, the surface and subsurface model are verified separately. The surface model is verified against the experimental data of surface runoff from a simulated rainfall. The surface model showed promising performance in terms of the reproduction of surface hydrograph. The overall discrepancy between the experimental and numerical model result in the reproduction of the hydrograph discharge is around 14%. For the subsurface flow model, the vertical moisture profile of the soil is verified against Philip’s analytical solution. The moisture profile obtained from the numerical model shows 100% agreement with Philip’s solution.

Soliton, periodic and superposition solutions to nonlocal (2+1)-dimensional, extended KdV equation derived from the ideal fluid model

We present new (2+1)-dimensional extended KdV (KdV2) equation derived within an ideal fluid model. Next, we show several families of analytic solutions to this equation. The solutions are expressed by functions of argument ξ=(kx+ly-ωt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi = (k x +l y-\\omega t)$$\\end{document}. We found the soliton solutions in the form Asech2(ξ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\,\ ext {sech}^{2}(\\xi )$$\\end{document}, periodic solutions in the form Acn2(ξ,m)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\,\ ext {cn}^{2}(\\xi ,m)$$\\end{document} and superposition solutions in the form A2[dn2(ξ,m)±mcn(ξ,m)dn(ξ,m)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{A}{2}[\ ext {dn}^{2}(\\xi ,m)\\pm \\sqrt{m}\\,\ ext {cn}(\\xi ,m)\ ext {dn} (\\xi ,m)]$$\\end{document} analogous to the solutions of (1+1)-dimensional, extended KdV equation and to the solutions to ordinary Korteweg-de Vries equation. On the other hand, the existence of these families of analytical solutions for the highly nonlinear non-local (2+1)-dimensional, extended KdV equation is astounding. The existence of essentially one-dimensional solutions to the (2+1)-dimensional extended KdV equation explains the enormous success of the one-dimensional nonlinear wave equations for the shallow water problem.

Constrained Receding Horizon Output Estimation of Linear Distributed Parameter Systems

We address constrained output estimation of discrete-time linear distributed parameter systems in the presence of plant and measurement disturbances. Sufficient conditions are proposed for the strong stability of the proposed moving horizon estimator. We further show that the discrete-time estimation results can be linked to continuous-time infinite-dimensional systems described by partial differential equations (PDEs) with unbounded disturbance and output operators by using Cayley-Tustin transformation. The theoretical results are illustrated with numerical examples on a one-dimensional (1D) wave equation and a 1D diffusion.

New Applications on the Flows of Space-like curves specified by Normal acceleration in Minkowski Space ℝ2,1

Abstract This paper investigates the kinematic motions of space-like and time-like curves specified by acceleration fields in Minkowski space ℝ2,1. Through the motion, the relationship between the acceleration fields and velocity fields is determined. In this study, we focus on studying the flows of inextensible space-like curves with a space-like principal normal vector specified by a normal acceleration that equals the curvature of the curve. Through the motion of the inextensible space-like curve with normal acceleration, we prove that the position vector of the curve satisfies a one-dimensional wave equation. We present some novel applications and visualize the flows of curves and their curvatures.

Dislocation detection of gas turbine materials using a nonlinear ultrasound modulation technique

Industrial gas turbines are used for generating electricity or driving other turbomachinery with the continuous development goal of further increasing machine efficiency. This is primarily achieved by raising the pressure ratio generated in the compressor and by increasing the turbine inlet temperature. Consequently, the hot gas components in gas turbines are subjected to extreme loads and the need for non-destructive testing and structural health monitoring techniques is becoming increasingly important to maintain these components. An important indicator for assessing the structural integrity is the determination of the initial plastic deformation.In this paper, a new method for the detection of plasticity was developed, which is based on a nonlinear ultrasonic two-frequency excitation. The one-dimensional wave equation was solved with a two-frequency excitation and combined with the expanded dislocation theory. As a result, various nonlinearity parameters were defined, showing a clear increasing or a decreasing behaviour with increasing plastic strain. This was experimentally proven with flat tensile specimen made of stainless steel and Inconel 718 (metal plates and additively manufactured). The new indicators allow the possibility to efficiently detect the initial plastic deformation in gas turbine components.

Hopf-like bifurcation in a wave equation at a removable singularity

It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and accumulates to zero. The upshot is that the singularity of the linearized operator at criticality which stems from the well known small divisor problem for the wave operator, is entirely removed without the need to exclude parameters via Diophantine conditions, nor the use of accelerated convergence schemes. Only the contraction mapping principle is used.