Search

Abstract In this paper, we investigate a system of coupled wave equations featuring boundary fractional damping applied to a portion of the domain. We first establish the well-posedness of the system, proving the existence and uniqueness of solutions through semi-group theory. While the system does not exhibit exponential stability, we demonstrate its strong stability. Furthermore, leveraging Arendt and Batty’s general criteria and certain geometric conditions, we prove a polynomial rate of energy decay for the solutions.

In this paper, we investigate a system of coupled wave equations featuring boundary fractional damping applied to a portion of the domain. We first establish the well-posedness of the system, proving the existence and uniqueness of solutions through semi-group theory. While the system does not exhibit exponential stability, we demonstrate its strong stability. Furthermore, leveraging Arendt and Batty's general criteria and certain geometric conditions, we prove a polynomial rate of energy decay for the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the deep numerical technique DeepNM, which is designed for solving one-dimensional (1D) hyperbolic conservation laws, particularly wave equations. By creatively integrating traditional numerical schemes with deep learning techniques, the method yields improvements over conventional approaches. Specifically, we compare this approach against two established classical numerical methods: the discontinuous Galerkin method (DG) and the Lax–Wendroff correction method (LWC). While maintaining a comparable level of accuracy, DeepNM significantly improves computational speed, surpassing conventional numerical methods in this aspect by more than tenfold, and reducing storage requirements by over 1000 times. Furthermore, DeepNM facilitates the utilization of higher-order numerical schemes and allows for an increased number of grid points, thereby enhancing precision. In contrast to the more prevalent PINN method, DeepNM optimally combines the strengths of conventional mathematical techniques with deep learning, resulting in heightened accuracy and expedited computations for solving partial differential equations. Notably, DeepNM introduces a novel research paradigm for numerical equation-solving that can be seamlessly integrated with various traditional numerical methods. [ABSTRACT FROM AUTHOR]

We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates for mild solutions using Lyapunov-type arguments. For the formulation of our results, we use the language of Hilbert complexes which provides all the tools required for our analysis and is also general enough to cover a number of interesting examples. Some of these are briefly discussed in the course of the manuscript. The functional analytic setting and the main arguments in our proofs are chosen such that they transfer almost verbatim to the discrete setting. We thus obtain corresponding decay results for numerical approximations of a variety of problems obtained by compatible discretization strategies which can be seen as our main contribution.

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]

This paper introduces a variety of approaches for solving 2D and 3D hyperbolic double interface problems. The methods are based on the Haar wavelet method, multiquadric radial basis function method, and integrated multiquadric radial basis function method. Temporal derivatives are handled using the second central difference and the Houbolt method. Various numerical approaches based on these methods are developed, and their implementations are discussed in complete detail. The paper evaluates and compares the performances of these approaches using both linear and nonlinear 2D and 3D double interface hyperbolic problems. Error analysis, conducted using the L-infinity norm, and efficiency assessments measured through CPU times contribute to a comprehensive understanding of the applicability and comparative effectiveness of the proposed methods. This study provides valuable insights for researchers and practitioners dealing with the challenges posed by interface problems in general.

In this study, we discuss a mathematical framework to handle the inverse problem for the applications of partial differential equations (PDEs). In particular, we focus on wave equations and attempt to identify the wave parameters such as wave velocity from scant measurements of the domain’s response to prescribed initial conditions. To this end, we need an algorithm to play the role of inverse PDE solver for full-waveform inversion. Over the past several years, multilayer neural networks have been developed and applied to a broad range of problems in applied mathematics and physics. Specifically, Physics Informed Neural Network (PINN) as a novel technique has been proposed recently for solving partial differential equations. In PINN’s algorithm, the mesh generation effort is not necessary as it is for any other numerical discretization method. The algorithm just needs a batch of points in which to apply the conditions set in the loss function. We employ PINN for solving the wave equations during the inversion process. The first objective of this research is to develop a robust and efficient algorithm based on PINN for the reconstruction of the wave velocity profile in heterogeneous media. Continuous and piecewise continuous functions are considered for the wave velocity target profiles. Next, we are interested in performing the inversion process of the wave equation in semi-infinite heterogeneous media which is one of the major advantages of the PINN in contrast to traditional numerical approaches. The last objective is to simultaneously recover the parameters of the viscously damped wave equation in heterogeneous domains. The effect of noisy measured response on the inversion process is also investigated.

In modern engineering, the dynamics of many practical problems can be described by hyperbolic distributed parameter systems. This paper is devoted to the adaptive prescribed performance control for a class of typical uncertain hyperbolic distributed parameter systems, since uncertainties are inevitable in practice. The systems in question simultaneously have unknown in-domain spatially varying damping coefficient and unknown boundary constant damping coefficient. Moreover, dynamic boundary condition is considered in the present paper. These characteristics make the control problem in the paper essentially different from those in the related works. To solve the problem, using adaptive technique based projection operator, backstepping method developed for ODEs and Lyapunov stability theories, a powerful adaptive prescribed performance control scheme is proposed to successfully guarantee that all states of the resulting closed-loop system are bounded, furthermore, the original system state converges to an arbitrary prescribed small neighborhood of the origin. Compared with the existing results, the developed control schemes can not only effectively handle the serious uncertainties, but also overcome the technical difficulties in the infinite-dimensional backstepping control design method caused by the dynamic boundary condition and guarantee prescribed performance.

Problems exhibiting wave-like characteristics pervade a diverse array of physical phenomena, including but not limited to longitudinal vibrations of elastic rods or beams, acoustic problems in fluid flow, electric signal transmission along cables, shock waves, chemical exchange processes in chromatography, sediment transport in rivers, plasma wave behavior, and the propagation of both electric and magnetic fields in the absence of charge and dielectric. This study introduces a novel series solution for wave-like problems, leveraging a newly developed technique within the Homotopy Perturbation Method (HPM). The proposed technique operates in the absence of a need for discretization, linearization, or restrictive assumptions, offering distinct advantages over conventional methods. This new version of HPM is designed to address wave-like problems. The technique rests on the assumption that the required solution can be represented as an infinite series sum. The proposed series demonstrates rapid convergence through the use of a control parameter. Initially, the parameter's scope is determined, followed by the selection of a single value that facilitates convergence. Various examples were applied using this technique, yielding satisfactory results. The novel version of HPM presented in this study hinges on the concept of representing the solution as an infinite series sum. This approach exhibits rapid convergence to the precise solution, enabled by the use of a control parameter. The parameter region is initially determined, followed by the selection of a single value that ensures convergence with a satisfactory sum count. [ABSTRACT FROM AUTHOR]

In acoustics, higher-order-in-time equations arise when taking into account a class of thermal relaxation laws in the modeling of sound wave propagation. In this work, we analyze initial boundary value problems for a family of such equations and determine the behavior of solutions as the relaxation time vanishes. In particular, we allow the leading term to be of fractional type. The studied model can be viewed as a gen-eralization of the well-established (fractional) Moore–Gibson–Thompson equation with three, in general nonlocal, convolution terms involving two different kernels. The interplay of these convolutions will influence the uniform analysis and the lim-iting procedure. To unify the theoretical treatment of this class of local and nonlocal higher-order equations, we relax the classical assumption on the leading-term kernel and consider it to be a Radon measure. After establishing uniform well-posedness with respect to the relaxation time of the considered general model, we connect it, through a delicate singular limit procedure, to fractional second-order models of linear acoustics. [ABSTRACT FROM AUTHOR]

We prove global Lipschitz stability for an inverse source problem of a system of wave equations on a Lorentzian manifold. The method used in this paper is widely known as the Bukhgeim–Klibanov method. However, the conventional method is not sufficient for the application to the hyperbolic partial differential equation with time-dependent coefficients to obtain the Lipschitz stability, and further innovations are needed. In this paper, we present an improved global Carleman estimate and an energy estimate to obtain the Lipschitz stability. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of the local existence for a solution to a nonlinear wave equation, with the dissipation given by a nonlinear form with the presence of a nonlinear memory term. Moreover, the global nonexistence of a solution is established using the test function method. We combine the Fourier transform and fractional derivative calculus to achieve our goal. [ABSTRACT FROM AUTHOR]

The spatial decay or growth behavior of a coupled nonlinear wave equation with damping and source terms is considered. By defining the wave equations in a cylinder or an exterior region, the spatial growth and decay estimates for the solutions are obtained by assuming that the boundary conditions satisfy certain conditions. We also show that the growth or decay rates are faster than those obtained by relevant literature. This kind of spatial behavior can be extended to a nonlinear system of viscoelastic type. In the case of decay, we also prove that the total energy can be bounded by known data.

A method of high-resolution acoustic tomography of scattering inhomogeneities based on the method of reverse time migration is proposed. An inhomogeneous background medium with a known distribution of refractive inhomogeneities is considered, where it is necessary to detect scattering inhomogeneities. Probing is carried out by broadband arbitrary waveforms. The method of reverse time migration is supplemented with linear filtering of the direct propagation field, which makes it possible to increase the resolution of reconstructed images. The results of numerical modeling and experiment are presented. Sounding was carried out in water at frequencies from 20 to 350 kHz using a signal with linear frequency modulation. The source of the ultrasonic signal was fixed and irradiated the objects under study in the water. The scattered signal was measured in a plane near the surface in water on a rectangular area with a step of less than 2 mm. Signal processing by the proposed method made it possible to reconstruct a three-dimensional image of scattering inhomogeneities with a resolution of about 7 mm. Comparison of the proposed method and the reverse time migration method showed that the proposed method allows obtaining a higher resolution.

This paper is concerned with the wave equation having a nonlinear distributed delay. First, we prove the local existence of the solutions by using the semigroup theory, where the source term is globally Lipschitz. Next, we establish the global existence of solutions and the energy decay result under the local Lipschitz source and suitable conditions on the initial data.

We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution.

In this work, we deal with the initial boundary value problem of solutions for a class of linear strongly damped nonlinear wave equations $ u_{tt}-\Delta u -\alpha \Delta u_t = f(u) $ in the frame of a family of potential wells. For this strongly damped wave equation, we not only prove the global-in-time existence of the solution, but we also improve the decay rate of the solution from the polynomial decay rate to the exponential decay rate.

We study a class of initial value problems for impulsive nonlinear wave equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. To prove our main results we give a suitable integral representation of the solutions of the considered problem. Then, we construct two operators so that any fixed point of their sum is a solution. [ABSTRACT FROM AUTHOR]

In this study, the analytical solution of the Schrödinger equation for the Trigonometric Inversely Quadratic plus Coulombic Hyperbolic Potential via the methodology of the supersymmetric approach was obtained. The energy equation and its corresponding wave functions were fully calculated. The theoretic quantities such as Shannon entropy and Fisher information were calculated using the normalized radial wave function. The results obtained for Shannon entropy satisfied Beckner, Bialynicki-Birula and Mycieslki (BBM) principle and Cramer Rao uncertainty inequality for Fisher information. These results are in excellent agreement with those in the literature. The result of our study goes against the observation pointed out by Okon et al. in their recent paper, who claimed that information entropic measures cannot be studied under Trigonometric Inversely Quadratic plus Coulombic Hyperbolic Potential.

The main goal of this work is to study a regularization method to reconstruct the solution of the backward non-linear hyperbolic equation $ u_{tt} + \alpha\Delta^2u_t +\beta \Delta ^2u = \mathcal{F}(x, t, u) $ come with the input data are blurred by random Gaussian white noise. We first prove that the considered problem is ill-posed (in the sense of Hadamard), i.e., the solution does not depend continuously on the data. Then we propose the Fourier truncation method for stabilizing the ill-posed problem. Base on some priori assumptions for the true solution we derive the error and a convergence rate between a mild solution and its regularized solutions. Also, a numerical example is provided to confirm the efficiency of theoretical results.

The engineering design of metamaterials with selected acoustic properties necessitates adequate prediction of the elastic wave propagation across various domains and specific frequency ranges. This study proposes a systematic approach centered on the finite element characterization of the three-dimensional Green's function for a representative volume element. The inherent characteristics of broadband waves and singular impulses contribute to notable challenges related to accuracy and high-frequency oscillations, and thus the emphasis is set on providing an exhaustive analysis for this numerical characterization scheme. The study focuses on the broadband wave dispersion and requisite considerations for numerical damping, and evaluates the impact of dissipation and space–time discretization schemes for optimal performance. In contrast to conventional methods that employ a plane wave, the proposed approach does not need extra assumptions on the enforcement of boundary conditions and can effectively consider the influences of length scale from the material configurations. A quasi-equiaxed polycrystalline ice microstructure is utilized as an application example for homogenizing heterogeneous materials, in line with advancements in cryo-ultrasonic testing techniques. • The approach combines FE and Green's function analysis to predict wave propagation. • Homogenization uses 3D assumption-free numerical Green functions over complex RVEs. • Key dissipation and discretization parameters for accuracy are reported. • The methodology is tested on polycrystalline ice microstructure. [ABSTRACT FROM AUTHOR]

The paper is devoted to the problem of the local existence for a solution to a nonlinear wave equation, with the dissipation given by a nonlinear form with the presence of a nonlinear memory term. Moreover, the global nonexistence of a solution is established using the test function method. We combine the Fourier transform and fractional derivative calculus to achieve our goal.

Abstract In this paper, we consider two internal stabilization problems for the multi-dimensional wave equation with a boundary time-delay. We prove that the first problem is well-posed in an appropriate functional space. Subsequently, we numerically study the exponential stability in a two-dimensional case under Geometric Control Condition (GCC) derived by Lebeau. In addition, we provide a numerical investigation of the second wave system, which corresponds to the two-dimensional variant of the system studied by Datko et al.

In this study, the analytical solution of the Schrödinger equation for the Trigonometric Inversely Quadratic plus Coulombic Hyperbolic Potential via the methodology of the supersymmetric approach was obtained. The energy equation and its corresponding wave functions were fully calculated. The theoretic quantities such as Shannon entropy and Fisher information were calculated using the normalized radial wave function. The results obtained for Shannon entropy satisfied Beckner, Bialynicki-Birula and Mycieslki (BBM) principle and Cramer Rao uncertainty inequality for Fisher information. These results are in excellent agreement with those in the literature. The result of our study goes against the observation pointed out by Okon et al. in their recent paper, who claimed that information entropic measures cannot be studied under Trigonometric Inversely Quadratic plus Coulombic Hyperbolic Potential. [ABSTRACT FROM AUTHOR]

In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

The purpose of this paper is to establish the existence and the uniqueness of a weak solution for the global functional energy associated to the wave equations coupled with the model of the von Karman evolution with rotational inertia and clamped boundary conditions. The resulting system of non-linear equations is solved by a numerical scheme based on the finite difference method. [ABSTRACT FROM AUTHOR]

In this paper, the Cauchy problem for nonlocal linear and nonlinear wave equations are studied. The equations include the general differential operators. The existence, uniqueness, L p -regularity properties and blow-up at finite time of solutions of the Cauchy problem is obtained. By choosing differential operators including in equations, the regularity properties of a different type wave equations are studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider two internal stabilization problems for the multi-dimensional wave equation with a boundary time-delay. We prove that the first problem is well-posed in an appropriate functional space. Subsequently, we numerically study the exponential stability in a two-dimensional case under Geometric Control Condition (GCC) derived by Lebeau. In addition, we provide a numerical investigation of the second wave system, which corresponds to the two-dimensional variant of the system studied by Datko et al. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the global exponential stability and the existence of periodic solutions of fuzzy wave equations are investigated. By variable substitution the system of partial differential equations (PDEs) is transformed from second order to first order. Some sufficient conditions that ensure the global exponential stability and the existence of periodic solution of the system are obtained by an analysis that uses a suitable Lyapunov functional. In addition, a concrete example is given to show the effectiveness of the results.

The solutions of a one-dimensional relativistic Klein-Gordon equation for a modified and deformed exponential-type potential are obtained using supersymmetric approach and parametric Nikiforov-Uvarov method respectively. The numerical values for lithium molecule were obtained using the energy equations from the two methods. The non-relativistic limit of the relativistic Klein-Gordn equation was obtained for each of the energy equation using a popular transformation invoked on the solution of the Klein-Gordon equation. Numerical values for lithium and sodium molecules were also generated based on the two methods. The generated values for both relativistic and non-relativistic were compared with the experimental values. These results agreed perfectly with the experimental values of the studied molecules.

A one-step new general mesh free scheme, which is based on radial basis functions, is presented for a viscous wave equation with variable coefficients. By constructing a simple extended radial basis function, it can be directly applied to wave propagation by using the strong form-based mesh free collocation method. There is no need to deal with the time-dependent variable particularly. Numerical results for a viscous wave equation with variable coefficients show that the proposed mesh free collocation method is simple with accurate solutions.

In this paper, fractional optimal control problem for two-dimensional coupled diffusion system with final observation is investigated. The fractional time derivative is considered in Atangana–Baleanu sense. Constraints on controls are imposed. First, by means of the classical control theory, the existence and uniqueness of the state for these systems is proved. Then, the necessary and sufficient optimality conditions for the fractional Dirichlet problems with the quadratic performance functional are derived. Finally we give some examples to illustrate the applicability of our results. The optimization problem presented in this paper constitutes a generalization of the optimal control problem of diffusion equations with Dirichlet boundary conditions considered in recent papers to coupled systems with Atangana–Baleanu time derivatives.

We consider a nonlinear Klein-Gordon system posed in an inhomogeneous medium Ω with smooth boundary ∂Ω subject to a local damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition, where the nonlinearities have not growth restrictions. We show that the energy of the system goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space. [ABSTRACT FROM AUTHOR]

Simulating and predicting wave propagation in porous media saturated with two fluids is an important issue in geophysical exploration studies. In this work, wave propagation in porous media with specified structures saturated with two immiscible fluids was studied, and the main objective was to establish a wave equation system with a relatively simple structure. The wave equations derived by Tuncay and Corapcioglu were analyzed first. It was found that the coefficient matrix of the equations tends to be singular due to the inclusion of a small parameter that characterizes the effect of capillary stiffening. Therefore, the previously established model consisting of three governing equations may be unstable under natural conditions. An improved model based on Tuncay and Corapcioglu’s work was proposed to ensure the nonsingularity of the coefficient matrix. By introducing an assumption in which one fluid was completely wrapped by the other, the governing equation of the wrapped fluid was degenerated. In this way, the coefficient matrix of wave equations became nonsingular. The dispersion and attenuation prediction resulting from the new model was compared with that of the original model. Numerical examples show that although the improved model consists of only two governing equations, it can obtain a result similar to that of the original model for the case of a porous medium containing gas and water, which simplifies the complexity of the calculations. However, in a porous medium with oil and water, the predictions of dispersion and attenuation produced by the original model obviously deviate from the normal trend. In contrast, the results of the improved model exhibit the correct trend with a smooth curve. This phenomenon shows the stability of the improved model and it could be used to describe wave propagation dispersions and attenuations of porous media containing two immiscible fluids in practical cases.

We obtain an asymptotic rate of decay for the radius of spatial analyticity of solutions to the nonlinear wave equation with initial data in the analytic Gevrey spaces. [ABSTRACT FROM AUTHOR]