Two-dimensional Nonlinear Wave Equation Research Articles

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>

Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons

In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional Kadomtsev–Petviashvili-I (KP-I) equation and spin-nonlinear Schrödinger (spin-NLS) equation via the deep neural networks learning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.

Investigate the dynamic nature of soliton solutions and bifurcation analysis to a new generalized two-dimensional nonlinear wave equation with its stability

In this current research, we investigate a new generalized two-dimensional nonlinear wave equation of engineering physics using two versatile approaches. Nonlinear wave models are very important in that they play a key role in modeling diverse occurrences in cosmology, solid mechanics and fluid dynamics, astrophysics, mathematical physics and engineering. The research work is based on three steps. Firstly, soliton solutions are secured via the modified Sardar sub-equation method (MSSEM) and the generalized exponential rational function method (GERFM). Secondly, for bifurcation analysis phase portraits of an equation are erected. Also, using the Galilean transformation, the given equation plane dynamical system is erected and all possible phase portraits are provided. Lastly, the modulation instability of the equation is examined. In order to better comprehend how the nonlinear wave equation depicts physical processes, the wave dynamics of the solutions are displayed to give the results a more physical perspective. The use of these findings in physics, engineering and complex media is essential.

Adding integrable defects to the Boussinesq equation

The purpose of this paper is to extend the store of models able to support integrable defects by investigating the two-dimensional Boussinesq nonlinear wave equation. As has been previously noted in many examples, insisting that a defect contributes to energy and momentum to ensure their conservation, despite the presence of discontinuities and the explicit breaking of translation invariance, leads to sewing conditions relating the two fields and their derivatives on either side of the defect. The manner in which several types of soliton solutions to the Boussinesq equation are affected by the defect is explored and reveals new effects that have not been observed in other integrable systems, such as the possibility of a soliton reflecting from a defect or of a defect decaying into one or two solitons.

Shock waves, periodic, topological kink and singular soliton solutions of a new generalized two dimensional nonlinear wave equation of engineering physics with applications in signal processing, electromagnetism and complex media

This article investigates a new generalized two-dimensional nonlinear wave equation of engineering physics with various applications in the fields of sciences and engineering. In this study, shock wave and solitary wave solutions were secured via the sine–Gordon technique. Moreover, various new group invariants along-side exact classical results of the equation were achieved through the utilization of Lie group theoretic techniques. Some of the solutions are gained with regards to Weierstrass functions, complex soliton, topological kink soliton as well as singular soliton. Besides, several algebraic and other solitary-wave-type solutions are obtained. Wave dynamics of the solutions are plotted to give more physical meanings to the obtained results and have a better knowledge of what the nonlinear wave equation represents in terms of physical phenomena. Application of the secured results in engineering (signal processing), physics (electromagnetism) and complex media are presented.

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.

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

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;

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

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

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.

Periodic solutions of two-dimensional wave equations with x-dependent coefficients and Sturm–Liouville boundary conditions

This paper considers the periodic solutions of a two-dimensional nonlinear wave equation with x-dependent coefficients subject to the Sturm–Liouville boundary conditions where x = (x 1, x 2) ∈ (0, π) × (0, π), and for i, j = 1, 2. Such a model arises from the forced vibrations of a nonhomogeneous membrane and the propagation of waves in nonhomogeneous media. By using the duality principles of Brézis (1983 Bull. Am. Math. Soc. 8 409–26), we first set up the theoretical framework and give an abstract theorem for the existence of periodic solution. Then, by analysing the spectral asymptotic behaviours of the weighted Sturm–Liouville problem and classifying the boundary conditions, we obtain the fundamental properties of the variable coefficients wave operator under appropriate assumptions for different class of boundary conditions. Finally, based on these properties and with the help of the abstract theorem, we establish some results on the existence and regularity of periodic solutions. To the best of our knowledge, the results are entirely new, and this is the first time to obtain such results for the two-dimensional nonlinear wave equation with x-dependent coefficients.

Efficient numerical algorithm with the second-order time accuracy for a two-dimensional nonlinear fourth-order fractional wave equation

In this article, we construct an efficient numerical algorithm with the second-order time accuracy for a two-dimensional nonlinear fourth-order fractional wave equation. We introduce an auxiliary variable to transform the fractional fourth-order wave problem into a low order coupled system, and then integrate the resulting equations by using Riemann–Liouville fractional integral. For the obtained fractional integral system, we use the Crank–Nicolson scheme in time with the weighted and shifted Grünwald integral (WSGI) approximation and the finite element method in space to arrive at the fully discrete coupled system. We show the detailed numerical algorithms and carry out numerical tests by taking two two-dimensional examples on rectangular and circular space domains to verify the effectiveness and convergence efficiency for our algorithm.

Error estimations of the fourth-order explicit Richardson extrapolation method for two-dimensional nonlinear coupled wave equations

Error estimations of the fourth-order explicit Richardson extrapolation method for two-dimensional nonlinear coupled wave equations

On the Parabolic and Hyperbolic Liouville Equations

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity lambda beta e^{beta u }, forced by an additive space-time white noise. (i) We first study SNLH for general lambda in {mathbb {R}}. By establishing higher moment bounds of the relevant Gaussian multiplicative chaos and exploiting the positivity of the Gaussian multiplicative chaos, we prove local well-posedness of SNLH for the range 0< beta ^2 < frac{8 pi }{3 + 2 sqrt{2}} simeq 1.37 pi . Our argument yields stability under the noise perturbation, thus improving Garban’s local well-posedness result (2020). (ii) In the defocusing case lambda >0, we exploit a certain sign-definite structure in the equation and the positivity of the Gaussian multiplicative chaos. This allows us to prove global well-posedness of SNLH for the range: 0< beta ^2 < 4pi . (iii) As for SdNLW in the defocusing case lambda > 0, we go beyond the Da Prato-Debussche argument and introduce a decomposition of the nonlinear component, allowing us to recover a sign-definite structure for a rough part of the unknown, while the other part enjoys a stronger smoothing property. As a result, we reduce SdNLW into a system of equations (as in the paracontrolled approach for the dynamical Phi ^4_3-model) and prove local well-posedness of SdNLW for the range: 0< beta ^2 < frac{32 - 16sqrt{3}}{5}pi simeq 0.86pi . This result (translated to the context of random data well-posedness for the deterministic nonlinear wave equation with an exponential nonlinearity) solves an open question posed by Sun and Tzvetkov (2020). (iv) When lambda > 0, these models formally preserve the associated Gibbs measures with the exponential nonlinearity. Under the same assumption on beta as in (ii) and (iii) above, we prove almost sure global well-posedness (in particular for SdNLW) and invariance of the Gibbs measures in both the parabolic and hyperbolic settings. (v) In Appendix, we present an argument for proving local well-posedness of SNLH for general lambda in {mathbb {R}}without using the positivity of the Gaussian multiplicative chaos. This proves local well-posedness of SNLH for the range 0< beta ^2 < frac{4}{3} pi simeq 1.33 pi , slightly smaller than that in (i), but provides Lipschitz continuity of the solution map in initial data as well as the noise.

The energy-preserving time high-order AVF compact finite difference scheme for nonlinear wave equations in two dimensions

In this paper, energy-preserving time high-order average vector field (AVF) compact finite difference scheme is proposed and analyzed for solving two-dimensional nonlinear wave equations including the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation. We first present the corresponding Hamiltonian system to the two-dimensional nonlinear wave equations, and further apply the compact finite difference (CFD) operator and AVF method to develop an energy conservative high-order scheme in two dimensions. The Lp-norm boundedness of two-dimensional numerical solution is obtained from the energy conservation property, which plays an important role in the analysis of the scheme for the two-dimensional nonlinear wave equations in which the nonlinear term satisfies local Lipschitz continuity condition. We prove that the proposed scheme is energy conservative and uniquely solvable. Furthermore, optimal error estimate for the developed scheme is derived for the nonlinear sine-Gordon equation and the nonlinear Klein-Gordon equation in two dimensions. Numerical experiments are carried out to confirm the theoretical findings and to show the performance of the proposed method for simulating the propagation of nonlinear waves in layered media.