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Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems. [ABSTRACT FROM AUTHOR]

We studied the longtime dynamical behavior of a wave equation with critical nonlinearity and nonlinear damping in an expanding domain. First, we employed the penalty approach to establish the local well-posedness of the weak solution. Subsequently, we illustrated energy dissipation and asymptotic compactness for both the penalized equations and the original equation. Lastly, we established the existence of pullback attractors for penalized equations and the original equation. [ABSTRACT FROM AUTHOR]

In this paper we study the fractal dimension of global attractors for a class of wave equations with (single-point) degenerate nonlocal damping. Both the equation and its linearization degenerate into linear wave equations at the degenerate point, and the usual approaches to calculate the dimension of the entirety of attractors do not work directly. Instead, we develop a new process concerning the dimension near the degenerate point individually and show the finite dimensionality of the attractor. [ABSTRACT FROM AUTHOR]

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]

This quest paper covers the most important topic in the field of mechanics. It is the nonlinear viscoelastic wave equations with variable degenerate damping and sources acting in both equations. We seek to prove that this system has no solution with positive initial energy. So, to establish our goal, we utilize the methods and techniques used by Said‐Houari in (J. Nonlinear. Differ. Equ. Appl, vol. 18, 2011, pp. 659‐684.). [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the robustness of the family of pullback attractors associated with a non-autonomous coupled system of strongly damped wave equations, which is a modified version of the well known Klein–Gordon–Zakharov system. Under appropriate hyperbolicity conditions, we establish the gradient-like structure of the limit pullback attractor associated with this evolution system, and we prove the continuity of the family of pullback attractors at zero. [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 consider the infinite-dimensional vector of frequencies , arising from a linear Klein – Gordon equation on the one-dimensional torus and prove that there exists a positive measure set of masses s for which satisfies a Diophantine condition similar to the one introduced by Bourgain in [14], in the context of the Schrödinger equation with convolution potential. The main difficulties we have to deal with are the asymptotically linear nature of the (infinitely many) s and the degeneracy coming from having only one parameter at disposal for their modulation. As an application we provide estimates on the inverse of the adjoint action of the associated quadratic Hamiltonian on homogenenous polynomials of any degree in Gevrey category. [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]

In the present study, we employed an analytical technique in order to determine the approximate/exact solutions to the distinct kind of partial differential equations (PDEs) (especially wave equation). Generalized homotopy perturbation method (GHPM) depends upon He's theory of homotopy perturbation and basic theory of least square method (LSM). The GHPM has been proven beneficial in obtaining the convergent solutions with ease. Applications of the scheme are demonstrated by assuming some examples. [ABSTRACT FROM AUTHOR]

This paper concern to determine a local energy decay estimate for synchronizable solutions of a coupled system of linear wave equations. Here, it is shown that the energy for such solutions decays at a rate of t−n$$ {t}^{-n} $$ (n$$ n $$ is the space dimension) when the coupling matrix satisfy the positive C1$$ {C}_1 $$‐compatibility condition. [ABSTRACT FROM AUTHOR]

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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 consider weak solutions for the obstacle-type viscoelastic ($ \nu>0 $) and very weak solutions for the obstacle inviscid ($ \nu = 0 $) Dirichlet problems for the heterogeneous and anisotropic wave equation in a fractional framework based on the Riesz fractional gradient $ D^s $ ($ 0

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]

Full waveform inversion (FWI) is a powerful tool for high-resolution subsurface parameter reconstruction. Due to the existence of local minimum traps, the success of the inversion process usually requires a good initial model. Our study primarily focuses on understanding the impact of source wavelets on the landscape of the corresponding optimization problem. We thus introduce a decomposition scheme that divides the inverse problem into two parts. The first step transforms the measured data into data associated with the desired source wavelet. Here, we consider inversions with known and unknown sources to mimic real scenarios. The second subproblem is the conventional FWI, which is much less dependent on an accurate initial model since the previous step improves the misfit landscape. A regularized deconvolution method and a convolutional neural network are employed to solve the source transformation problem. Numerical experiments on the benchmark models demonstrate that our approach improves the gradient's quality in the subsequent FWI and provides a better inversion performance. [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]

In this work, we propose a novel preconditioned Krylov subspace method for solving an optimal control problem of wave equations, after explicitly identifying the asymptotic spectral distribution of the involved sequence of linear coefficient matrices from the optimal control problem. Namely, we first show that the all-at-once system stemming from the wave control problem is associated to a structured coefficient matrix-sequence possessing an eigenvalue distribution. Then, based on such a spectral distribution of which the symbol is explicitly identified, we develop an ideal preconditioner and two parallel-in-time preconditioners for the saddle point system composed of two block Toeplitz matrices. For the ideal preconditioner, we show that the eigenvalues of the preconditioned matrix-sequence all belong to the set (-3/2, -1/2) ∪ (1/2,3/2) well separated from zero, leading to mesh-independent convergence when the minimal residual method is employed. The proposed parallel-in-time preconditioners can be implemented efficiently using fast Fourier transforms or discrete sine transforms, and their effectiveness is theoretically shown in the sense that the eigenvalues of the preconditioned matrix-sequences are clustered around ±1, which leads to rapid convergence. When these parallel-in-time preconditioners are not fastly diagonalizable, we further propose modified versions which can be efficiently inverted. Several numerical examples are reported to verify our derived localization and spectral distribution result and to support the effectiveness of our proposed preconditioners and the related advantages with respect to the relevant literature. [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.

This paper proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. The comparison results demonstrate that our model is ten times more accurate than ADCP. Our formulation is capable of simulating a variety of practical measurement scenarios. [ABSTRACT FROM AUTHOR]

We develop a hybrid spatial discretization for the wave equation in second order form, based on high-order accurate finite difference methods and discontinuous Galerkin methods. The hybridization combines computational efficiency of finite difference methods on Cartesian grids and geometrical flexibility of discontinuous Galerkin methods on unstructured meshes. The two spatial discretizations are coupled with a penalty technique at the interface such that the overall semidiscretization satisfies a discrete energy estimate to ensure stability. In addition, optimal convergence is obtained in the sense that when combining a fourth order finite difference method with a discontinuous Galerkin method using third order local polynomials, the overall convergence rate is fourth order. Furthermore, we use a novel approach to derive an error estimate for the semidiscretization by combining the energy method and the normal mode analysis for a corresponding one-dimensional model problem. The stability and accuracy analysis are verified in numerical experiments. [ABSTRACT FROM AUTHOR]

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 first show that under a suitable balanced repartition of the mixed controls within the system, Kalman's rank condition is still necessary and sufficient for the uniqueness of solution to the adjoint system associated with incomplete internal and boundary observations, therefore for the approximate controllability of the primary system by means of mixed controls. Then we study the stability of the approximately synchronizable state by groups with respect to applied controls. [ABSTRACT FROM AUTHOR]

This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain $\Omega = \mathbb {T}\times \mathbb {R}$ with $\mathbb {T}=[0,\,1]$ being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field $(1,\,0)$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in $H^{1}$ as $t\to \infty$. As a consequence, the solution converges to its horizontal average asymptotically. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with a viscoelastic wave equation with infinite memory and nonlinear frictional damping of variable-exponent type. First, we establish explicit and general decay results with a very general assumption on the relaxation function. Then, we remove the constraint imposed on the boundedness condition on the initial data used in the earlier results in the literature. Finally, we perform several numerical tests to illustrate our theoretical findings. This study generalizes and improves previous literature outcomes. [ABSTRACT FROM AUTHOR]

The paper is devoted to the boundary controllability of the abstract wave equations when the control is exerted on a part of the boundary by means of one control. We give a Kalman-type condition and give a description of the attainable set. The equation includes a linear operator A defined in a Hilbert space H, in which by choosing H and A, we can obtain boundary controllability properties of numerous classes of nonlocal mixed value problems for wave equations which occur in a wide variety of physical systems. In this respect, we derived boundary controllability properties of the mixed problem for infinite many systems of wave equations, nonlocal mixed problem for degenerate wave equations and for high-order wave equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization of a coupled wave system formed by one localized nonregular fractional viscoelastic damping of Kelvin–Voigt type and localized nonsmooth coefficients. Our main aim is to prove that the C0$$ {C}_0 $$‐semigroup associated with this model is strong stability and decays polynomially at a rate of t−1$$ {t}^{-1} $$. By introducing a new system to deal with fractional Kelvin–Voigt damping, we obtain a new equivalent augmented system, so as to show the well‐posedness of the system based on Lumer–Phillips theorem. We achieve the strong stability for the C0$$ {C}_0 $$‐semigroup associated with this new model by using a general criteria of Arendt–Batty and then turn out a polynomial energy decay rate of order t−1$$ {t}^{-1} $$ with the help of a frequency domain approach. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the integral type problem for linear and nonlinear abstract wave equations with operator coefficient in E-valued L p , 1 ≤ p ≤ ∞ spaces. The equation includes an abstract linear operator A that defined in a Banach space E. Here, the existence, uniqueness, uniform estimates of solution and quality properties of integral wave problem are derived. We also obtain the same properties for solutions of mixed integral problem for nonlocal and anisotropic classes, by choosing the space E and operator A. [ABSTRACT FROM AUTHOR]

Given a densely defined skew-symmetric operator A 0 on a real or complex Hilbert space V, we parameterize all m-dissipative extensions in terms of contractions Φ : H - → H + , where H - and H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0 -group if and only if Φ is a unitary operator. As a corollary we obtain the parametrization of all selfadjoint extensions of a symmetric operator by unitary operators from H - to H + . Our results extend the theory of boundary triples initiated by von Neumann and developed by V. I. and M. L. Gorbachuk, J. Behrndt and M. Langer, S. A. Wegner and many others, in the sense that a boundary quadruple always exists (even if the defect indices are different in the symmetric case). [ABSTRACT FROM AUTHOR]

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.

Backscattered wave acoustic tomography using wideband probing signals makes it possible to obtain three-dimensional (3D) images of scattering inhomogeneities. Signal processing based on the reverse time migration (RTM) method allows one to take into account the influence of background refractive obstacles of the medium to minimize distortions of reconstructed tomographic images. We propose a noniterative method of acoustic tomography in an immersion medium based on RTM approach supplemented with linear signal preprocessing to enhance resolution of reconstructing tomography images. The visualization of scattering objects is based on wave inversion from the measurement area considering the probing wave field specially distorted to perform regularized back convolution. The applicability of the proposed method for visualizing scattering objects in water is shown analytically, numerically and experimentally. The proposed method is resistant to noise according to regularization. The results obtained show the agreement between the numerical and analytical solution. Using the example of sounding with linear frequency modulation signals, it is demonstrated that the proposed method allows increasing the resolution of tomographic images in comparison with conventional RTM. The novelty of the proposed method is the preliminary filtration of the forward propagation wave in the course of solving the inverse problem. This approach improves the resolution of tomographic images and allows considering the influence of obstacles. [ABSTRACT FROM AUTHOR]

We consider a model of nonlinear wave equations with periodically varying wave speed and periodic external forcing. By imposing non-resonance conditions on the frequency, we establish the existence of response solutions (i.e., periodic solutions with the same frequency as the forcing) for such a model in a Cantor set of asymptotically full measure. The proof relies on a Lyapunov–Schmidt reduction together with the Nash–Moser iteration. [ABSTRACT FROM AUTHOR]

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.