Adding integrable defects to the Boussinesq equation

A Conservative Spectral Method for Several Two-Dimensional Nonlinear Wave Equations

<abstract><p>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.</p></abstract>

We develop a technique making it possible to handle the problem of separation of variables in nonlinear differential equations. Using it we obtain a number of new two-dimensional nonlinear wave equations admitting separation of variables and construct their exact solutions.

Analysis of modified Godunov type schemes for the two-dimensional linear wave equation with Coriolis source term on cartesian meshes

A widely convergent generalized pulse-spectrum technique for the inversion of two-dimensional acoustic wave equation

The Galerkin solution of two-dimensional TE wave equation is derived from the two-dimensional vector wave equation by using basis (interpolation) functions defined in each element. And then, considering with the first-order absorbing boundary condition the time-domain recursive formulation is obtained by using Newmark-β method . Finally, a numerical example is given to illustrate the feasibility of the presented scheme.

Purpose Semilinear wave equations with different source terms describe acoustic wave motion in fluids, shock wave formation that decelerates fluid from supersonic to subsonic speeds and quenching phenomena in micro-electro mechanical systems devices with fluid mechanical applications. This paper aims to investigate the quenching behavior of numerical solutions for a two-dimensional semilinear wave equation with an inverse power law term. Design/methodology/approach The localized radial basis function-generated finite difference (RBF-FD) method is used for approximating numerical solutions in space, and the finite difference scheme is used for temporal discretization. A discrete energy analysis is conducted to evaluate the local stability of the developed numerical scheme. Findings The energy functional of the classical solution is defined. The numerical results demonstrate finite-time quenching, and the influence of various parameters is assessed through detailed numerical simulation. Originality/value An RBF-FD approach is applied to confront the quenching phenomena in one- and two-dimensional cases. Stability and the computational performance of the proposed numerical scheme are verified numerically. The impact of various parameters and domains on quenching time is studied in detail.

In 1963, Kumano-go presented one non-uniqueness example for the two-dimensional wave equation with a time-dependent potential. We construct infinitely many non-uniqueness examples with different wave numbers at infinity for Cauchy problems of the two-dimensional wave equation and Schrödinger equation as a generalization of the construction by Kumano-go.

The Cauchy problem is considered for the two-dimensional wave equation with velocity $c=\sqrt x_1$ on the half-plane $\{x_1\geq 0,x_2\}$, with initial data localized in a neighborhood of the point $(1,0)$. This problem serves as a model problem in the theory of beach run-up of long small-amplitude surface waves excited by a spatially localized instantaneous source. The asymptotic expansion of the solution is constructed with respect to a small parameter equal to the ratio of the source linear size to the distance from the $x_2$-axis (the shoreline). The construction involves Maslov’s canonical operator modified to cover the case of localized initial conditions. The relationship of the solution with the geometrical optics ray diagram corresponding to the problem is analyzed. The behavior of the solution near the $x_2$-axis is studied. Simple solution formulas are written out for special initial data.

The characteristics of unsteady flow motions have long been a serious concern in the study of various fluid dynamic and combustion problems. With the advancement of computer resources, numerical approaches to these problems appear to be feasible. The objective of this paper is to assess the accuracy of several numerical schemes for unsteady flow calculations. In the present study, Fourier error analysis is performed for various numerical schemes based on a two-dimensional wave equation. Four methods sieved from the error analysis are then adopted for further assessment. Model problems include unsteady quasi-one-dimensional inviscid flows, two-dimensional wave propagations, and unsteady two-dimensional inviscid flows. According to the comparison between numerical and exact solutions, although second-order upwind scheme captures the unsteady flow and wave motions quite well, it is relatively more dissipative than sixth-order central difference scheme. Among various numerical approaches tested in this paper, the best performed one is Runge-Kutta method for time integration and six-order central difference for spatial discretization.

Type-II hidden symmetries of the linear, two-dimensional and three-dimensional wave equations are analysed. These hidden symmetries are Lie point symmetries that appear in addition to the inherited point symmetries when the number of independent and dependent variables of a partial differential equation is reduced by a Lie point symmetry. The provenance of these hidden symmetries of partial differential equations is identified to be the same as found recently for some nonlinear partial differential equations. The appearance of Type-II hidden symmetries depends not only on the Lie symmetries used but on the order in which the symmetries are applied. The presence of Type-II hidden symmetries of partial differential equations complicates the prediction of symmetry reductions based on the Lie algebra associated with the original Lie point symmetries.

For the conservative and non-conservative schemes of nonlinear evolution equations, by taking the two-dimensional shallow water wave equations as an example, a comparative analysis on computational stability is carried out. The relationship between the nonlinear computational stability, the structure of the difference schemes, and the form of initial values is also discussed.

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian Gr_kn. More recently several authors have studied the regular solutions that one obtains in this way: these come from points of the totally non-negative part of the Grassmannian (Gr_kn)_{>= 0}. In this paper we exhibit a surprising connection between the theory of total positivity for the Grassmannian, and the structure of regular soliton solutions to the KP equation. By exploiting this connection, we obtain new insights into the structure of KP solitons, as well as new interpretations of the combinatorial objects indexing cells of (Gr_kn)_{>= 0}. In particular, we completely classify the spatial patterns of the soliton solutions coming from (Gr_2n)_{>0}, as well as those coming from (Gr_kn)_{>= 0} when the absolute value of the time parameter is sufficiently large. We also demonstrate an intriguing connection between soliton graphs for (Gr_kn)_{>0} and the cluster algebras of Fomin and Zelevinsky, and we use this connection to solve the inverse problem for generic KP solitons coming from (Gr_kn)_{>0}. Finally we construct all the soliton graphs for (Gr_2n)_{>0} using the triangulations of n-gon.

Energy conserving Galerkin approximation of two dimensional wave equations with random coefficients

A first, we used our knowledge of Fourier series to solve several interesting boundary value problems by the method of separation of variables. The success of our method depended to a large extent on the fact that the domains under consideration were easily described in Cartesian coordinates. In this paper/research we address problems where the domains are easly described in polar and cylindrical coordinates. Spesifically we consider boundary value problems for the wave, heat, Laplace and Poisson equation over disks or cylinders. Upon restating these problems in suitable coordinat systems and separating variables, we will encounter new ordinary differential equations, Bessel’s equation, whose solutions are called Bessel function in ways analogous to Fourier series expansions. The vibrations of the membrane are governed by the two-dimensional wave equation, which will be expressed in polar coordinantes, because these are the coordinates best suited to this problem. Finally, we will solve the two dimensional wave equation in polar coordinates (general case).