Two-dimensional Wave Equation Research Articles

Global well-posedness of the two-dimensional stochastic viscous nonlinear wave equations

We study well-posedness of viscous nonlinear wave equations (vNLW) on the two-dimensional torus with a stochastic forcing. In particular, we prove pathwise global well-posedness of the stochastic defocusing vNLW with an additive stochastic forcing Dαξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^\\alpha \\xi $$\\end{document}, where α<12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha < \\frac{1}{2}$$\\end{document} and ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\xi $$\\end{document} denotes the space–time white noise.

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations

Abstract The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs), to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in Du et al. (2018, Commun. Comput. Phys., 24, 1049–1072) and Du et al. (2018, SIAM J. Sci. Comp., 40, A1430–A1445) to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green’s first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

Hbox {U}^p-Net: a generic deep learning-based time stepper for parameterized spatio-temporal dynamics

In the age of big data availability, data-driven techniques have been proposed recently to compute the time evolution of spatio-temporal dynamics. Depending on the required a priori knowledge about the underlying processes, a spectrum of black-box end-to-end learning approaches, physics-informed neural networks, and data-informed discrepancy modeling approaches can be identified. In this work, we propose a purely data-driven approach that uses fully convolutional neural networks to learn spatio-temporal dynamics directly from parameterized datasets of linear spatio-temporal processes. The parameterization allows for data fusion of field quantities, domain shapes, and boundary conditions in the proposed U^p-Net architecture. Multi-domain U^p-Net models, therefore, can generalize to different scenes, initial conditions, domain shapes, and domain sizes without requiring re-training or physical priors. Numerical experiments conducted on a universal and two-dimensional wave equation and the transient heat equation for validation purposes show that the proposed U^p-Net outperforms classical U-Net and conventional encoder–decoder architectures of the same complexity. Owing to the scene parameterization, the U^p-Net models learn to predict refraction and reflections arising from domain inhomogeneities and boundaries. Generalization properties of the model outside the physical training parameter distributions and for unseen domain shapes are analyzed. The deep learning flow map models are employed for long-term predictions in a recursive time-stepping scheme, indicating the potential for data-driven forecasting tasks. This work is accompanied by an open-sourced code.

Chaotic vibration of a two-dimensional wave equation with nonlinear boundary condition

In this paper, we consider the chaotic dynamical behavior of a two-dimensional wave equation due to an energy-injecting boundary condition and an energy dissipation boundary condition. By using the chaotic mapping theory and the method of characteristic, we prove the onset of chaos in the sense of exponential growth of total variation of the 2D wave equation. Moreover, We also prove that the system has not the chaotic oscillations when the parameters enter certain ranges. Numerical examples are provided to verify the effectiveness of our theoretical results.

Particle trajectories in a weakly nonlinear long-wave model on a shear flow

This work centers on the numerical examination of particle trajectories associated with the propagation of long water waves of small but finite amplitude on a background shear flow over a flat bottom. Taking into consideration the assumption that the nonlinear and dispersive effects are small and of the same magnitude, the Boussinesq-type equations for two-dimensional water waves on a background flow with constant vorticity are derived. Restricting attention to waves propagating in a single direction, we find a new version of the Benjamin-Bona-Mahony (BBM) equation which takes into account the effect of vorticity. In order to investigate the particle trajectories of the flow, an approximate velocity field associated with the derivation of the BBM equation over a shear flow is obtained. Several cases of particle paths under surface waves profiles such as solitary waves and periodic traveling waves are examined.

Approximate solutions to shallow water wave equations by the homotopy perturbation method coupled with Mohand transform

In this paper, the Mohand transform-based homotopy perturbation method is proposed to solve two-dimensional linear and non-linear shallow water wave equations. This approach has been proved suitable for a broad variety of non-linear differential equations in science and engineering. The variation trend of the water surface elevation at different time levels and depths are given by some graphs. Moreover, the obtained solutions are compared with the existing results, which show higher efficiency and fewer computations than other approaches studied in the literature.

A Comparison of the Adomian Decomposition Method and Variational Iteration Method for a Two-Dimensional Nonlinear Wave Equation

This paper studies a two-dimensional wave equation in the presence of power and derivative nonlinearity subject to suitable initial conditions. It aims to construct the exact-analytical solution of a two-dimensional nonlinear wave equation using two different semi-analytical methods and to compare the obtained results. First, a well-known Adomian decomposition method (ADM) based on operators is employed. Many researchers use the ADM to investigate several scientific applications, and this method straightforwardly attacks the problem without using linearization, discretization, perturbation, or any other restrictive assumption that may change the physical behavior of the problem. Second, the variational iteration method (VIM) also provides rapid convergent successive approximations of the closed-form solutions if it exists; otherwise, it provides an approximation of a high degree accuracy level even in case of few obtained iterations. The obtained results are drawn graphically and presented in the tables. Both methods provide almost the same solutions in each nonlinearity case, and VIM has computational advantages over ADM in computation size. Further, ADM and VIM provide a series of solutions that converge in a very small time domain, which limits these methods. Keywords: Adomian decomposition method, variational iteration method, approximations, partial differential equations, two-dimensional wave equation. https://doi.org/10.55463/issn.1674-2974.50.2.3

An optimization method for the inverse scattering problem of the biharmonic wave

We consider an inverse scattering problem of reconstructing the shape of a cavity in an infinitely thin plate, where the out-of-plane displacement is governed by the two-dimensional biharmonic wave equation. We first approximate the out-of-plane displacement of the plate by potentials over an auxiliary circle and then reformulate the inverse problem into an optimization problem. The convergence properties are established to show that when the regularization parameter tends to zero, the minimizer of the optimization problem tends to a solution of the inverse scattering problem. Numerical experiments are designed to verify the performance of the proposed method.

Explicit Richardson extrapolation methods and their analyses for solving two-dimensional nonlinear wave equation with delays

&lt;abstract&gt;&lt;p&gt;In this study, we construct two explicit finite difference methods (EFDMs) for nonlinear wave equation with delay. The first EFDM is developed by modifying the standard second-order EFDM used to solve linear second-order wave equations, of which stable requirement is accepted. The second EFDM is devised for nonlinear wave equation with delay by extending the famous Du Fort-Frankel scheme initially applied to solve linear parabolic equation. The error estimations of these two EFDMs are given by applying the discrete energy methods. Besides, Richardson extrapolation methods (REMs), which are used along with them, are established to improve the convergent rates of the numerical solutions. Finally, numerical results confirm the accuracies of the algorithms and the correctness of theoretical findings. There are few studies on numerical solutions of wave equations with delay by Du Fort-Frankel-type scheme. Therefore, a main contribution of this study is that Du Fort-Frankel scheme and a corresponding new REM are constructed to solve nonlinear wave equation with delay, efficiently.&lt;/p&gt;&lt;/abstract&gt;

High-order compact difference methods for solving two-dimensional nonlinear wave equations

&lt;abstract&gt;&lt;p&gt;Nonlinear wave equations are widely used in many areas of science and engineering. This paper proposes two high-order compact (HOC) difference schemes with convergence orders of $ O\left({{\tau ^4} + h_x^4 + h_y^4} \right) $ that can be used to solve nonlinear wave equations. The first scheme is a nonlinear compact difference scheme with three temporal levels. After calculating the second-order spatial derivatives of the previous time level using the Padé scheme, numerical solutions of the next time level are obtained through repeated iterations. The second scheme is a three-level linearized compact difference scheme. Unlike the first scheme, iterations are not required and it obtains numerical solutions through an explicit calculation. The two proposed schemes are applied to solutions of the coupled sine-Gordon equations. Finally, some numerical experiments are presented to confirm the effectiveness and accuracy of the proposed schemes.&lt;/p&gt;&lt;/abstract&gt;

Note on the global wellposedness of two-dimensional incompressible Navier–Stokes fluid structure interaction model

In this note we study the global wellposedness of a two-dimensional incompressible Navier–Stokes fluid structure interaction(FSI) model. The existence and uniqueness of the global strong and smooth solution to the initial boundary value problem for one class of the FSI model in the two-dimensional case, which is one model of two-dimensional incompressible Navier–Stokes equations coupled by the two-dimensional wave equation by the suitable boundary conditions at the interface boundary, are obtained for any smooth initial data under the assumption of the suitable compatibility conditions on the initial data at the interface boundary and external boundary of the domain.

Symmetries and solutions for the inviscid oceanic Rossby wave equation

The -and -dimensional inviscid Rossby wave equations are analysed using Lie symmetry techniques. The travelling-wave reductions for the equation lead to a third-order ordinary differential equation from which the propagation properties are derived. It is observed that the wave has easterly phase velocity and westerly group velocity. Also, the wave propagates slightly faster in the f-plane than the β-plane. The dispersion relation derived from the third-order equation shows that the Rossby equation transports energy both eastward and westward and its speed is reduced successively and the propagation remains eastward. As for the two-dimensional Rossby wave equation, certain solutions which behave like solitary waves after a certain time are plotted. Interestingly, for certain symmetries the reductions lead to the Riccati's, Abel's, Euler's type and to some linearized equations. Moreover, certain reductions lead to highly nonlinear equations which are analysed by the singularity analysis method.

Two nonnegative solutions for two-dimensional nonlinear wave equations

We study a class of initial value problems for two-dimensional nonlinear wave equations. A new topological approach is applied to prove the existence of at least two nonnegative classical solutions. The arguments are based upon a recent theoretical result.

On the two-dimensional singular stochastic viscous nonlinear wave equations

On the two-dimensional singular stochastic viscous nonlinear wave equations

About complex intelligent technologies for techno-ecological events control in the water area

Aspects of the important task solution of creating complex intelligent decision-making support technologies for the identification of techno-ecological event (TEE) and the optimal selection of the sequence of available measures to reduce the life cycle of this TEE in the water area in order to minimize material losses are considered ("CONTROL_TEE" system). In the V.M. Glushkov Institute of Cybernetics of NAS of Ukraine, Concern "BaltRobotics" (Ukraine-Poland), NTU of Ukraine "Igor Sikorsky KPI" study of the possibility of theoretical development, research and practical implementation of methods and tools that make up the information technology of research design (informational, mathematical, algorithmic, software, technical, organizational support) of robots intended for reconnaissance and neutralization of TEE in a number of environment. For the classifying waves, mathematical models of the propagation of both running and standing waves in the sea area were obtained and solved. The structure of the information storage of the situation center has been developed. In order to create a database wave classification and mathematical and computer modeling were carried out. The deterministic process of sound propagation in a flat waveguide in the homogeneous mode is considered. Special boundary value problems and Cauchy problems are solved for the two-dimensional wave equation and, accordingly, for the Helmholtz equation. Calculation formulas for sound pressure and corresponding to its velocities are obtained in an analytical closed form. In the general case, the tangent and normal components of the velocity vector and the hydrodynamic potential are calculated in the form of Fourier series by the methodology of the works of Bilonosov, Ovsienko, Li, Zinchenko, Zinchenko, Nogin.

Efficient sixth-order finite difference method for the two-dimensional nonlinear wave equation with variable coefficient

Efficient sixth-order finite difference method for the two-dimensional nonlinear wave equation with variable coefficient

A class of cylindrically symmetric exact solutions of the wave equation

A class of analytically exact solutions is derived of the two-dimensional wave equation. The solutions are due to a point source (i.e. a uniform line source in 3D) at the origin r=0 with polynomial time dependence that starts at t=0. They are essentially similarity solutions in the variable c0t/r. Their explicit formulation is supported by analytical results for large t and small r behaviour. By taking advantage of linearity, sources of finite duration can be modelled by subtracting from a solution its delayed version. For illustration, some examples are given graphically as functions of r, of t, and of t−r/c0, for a number of points in time and location, respectively.

An efficient numerical simulation of the two-dimensional semilinear wave equation

An efficient numerical simulation of the two-dimensional semilinear wave equation

Approximate Analytical Solution of Two-Dimensional Nonlinear Time-Fractional Damped Wave Equation in the Caputo Fractional Derivative Operator

In this work, we proposed a new method called Laplace–Padé–Caputo fractional reduced differential transform method (LPCFRDTM) for solving a two-dimensional nonlinear time-fractional damped wave equation subject to the appropriate initial conditions arising in various physical models. LPCFRDTM is the amalgamation of the Laplace transform method (LTM), Padé approximant, and the well-known reduced differential transform method (RDTM) in the Caputo fractional derivative senses. First, the solution to the problem is gained in the convergent power series form with the help of the Caputo fractional-reduced differential transform method. Then, the Laplace–Padé approximant is applied to enlarge the domain of convergence. The advantage of this method is that it solves equations simply and directly without requiring enormous amounts of computational work, perturbations, or linearization, and it expands the convergence domain, leading to the exact answer. To confirm the effectiveness, accuracy, and convergence of the proposed method, four test-modeling problems from mathematical physics nonlinear wave equations are considered. The findings and results showed that the proposed approach may be utilized to solve comparable wave equations with nonlinear damping and source components and to forecast and enrich the internal mechanism of nonlinearity in nonlinear dynamic events.

Study of the Fragile Points Method for solving two-dimensional linear and nonlinear wave equations on complex and cracked domains

This paper presents the meshless Fragile Points Method (FPM) for the two-dimensional linear and nonlinear wave equations on irregular and complex domains. Time derivatives of the wave equation are discretized using finite difference schemes, and the theorems of stability and convergence of this semi-discrete scheme are given. A simple Galerkin was applied with the domain divided into subdomains by a Voronoi diagram for spatial discretization of problems. Numerical flux corrections are applied to avoid incompatibilities in this study. In case of encountering a crack or discontinuity in the common neighborhood of the selected points, the connection between the two points is cut off without making any particular change in the result. Hence in this method, the points are called “fragile” and it is suitable for solving problems with cracks and discontinuities. A predictor–corrector scheme is used to obtain an appropriate approximation for the nonlinear term in the wave equation. FPM yields symmetric and sparse point stiffness matrices, making it a computationally efficient method that reaches the appropriate accuracy in a short time according to the conditions of the problem. Test problems on complex and cracked domains are evaluated that show the acceptable accuracy and efficiency of the proposed method.