The convergence of fully discrete Galerkin approximations of the Rosenau equation

In this paper, we shall analyze the fully discrete Galerkin type approximations to solutions of the Rosenau equation. We provide the numerical results of several cases.     

The convergence of finite element Galerkin solution for the Roseneau equation

In this paper, we analyze the convergence of the semidiscrete solution of the Roseneau equation. We introduce the auxiliary projection of the solution, and derive the optimal convergence of the semidiscrete solution as well as the auxiliary projection inL ² normed space.     

The projection Galerkin method for solving the time-independent differential diffusion equation in a semi-infinite domain

Using the diffusion equation as an example, results of applying the projection Galerkin method for solving time-independent heat and mass transfer equations in a semi-infinite domain are presented. The convergence of the residual corresponding to the approximate solution of the timeindependent diffusion equation obtained by the projection method using the modified Laguerre functions is proved. Computational results for a two-dimensional toy problem are presented.     

Radius of analyticity and exponential convergence for spectral projections of the generalized KdV equation

In this paper an exponential convergence rate for a spectral projection of the periodic initial-value problem for the generalized KdV equation is proved. Based on this convergence result, a method for determining the radius of analyticity of solutions of the generalized KdV equation is derived. Results from the new method and a similar method are compared.     

Weighted estimate for the convergence rate of a projection difference scheme for a quasilinear parabolic equation

A novel technique is proposed for analyzing the convergence of a projection difference scheme as applied to the initial value problem for a quasilinear parabolic operator-differential equation with initial data u ₀ ∈ H. The technique is based on the smoothing property of solutions to the differential problem for t > 0. Under certain conditions on the nonlinear term, a new estimate of order \(O(\sqrt \tau + h)\) for the convergence rate in a weighted energy norm is obtained without using a priori assumptions on the additional smoothness of weak solutions.     

The general analytical and numerical solution for the modified KdV equation with convergence analysis

We investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge–Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge–Kutta discontinuous Galerkin method by some numerical illustrations. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability and convergence of the spectral Galerkin method for the Cahn‐Hilliard equation

A spectral Galerkin method in the spatial discretization is analyzed to solve the Cahn‐Hilliard equation. Existence, uniqueness, and stabilities for both the exact solution and the approximate solution are given. Using the theory and technique of a priori estimate for the partial differential equation, we obtained the convergence of the spectral Galerkin method and the error estimate between the approximate solution u_N(t) and the exact solution u(t). © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.     

Symmetry reductions for a dissipation-modified KdV equation

In this paper we give a group classification for a dissipation-modified Korteweg-de Vries equation by means of the Lie method of the infinitesimals. We prove that, by using the nonclassical method, we get several new solutions which are unobtainable by Lie classical symmetries. We obtain nonclassical symmetries that reduce the dissipation-modified Korteweg-de Vries equation to ordinary equations with the Painlevé property. These solutions have not been derived elsewhere by the singular manifold method.     

Computing solutions to a forced KdV equation

In this paper a special forced Korteweg–de Vries (KdV) equation is considered. This equation is established by recent studies as a simple mathematical model of describing the physics of a shallow layer of fluid subject to external forcing. It serves as an analytical model of tsunami generation by submarine landslides. The bilinear form for this equation is obtained with the aid of Hirota’s method. Some of its one-, two- and three-soliton as well as breather-type soliton solutions and other interesting solutions are derived.     

Galerkin and weighted Galerkin methods for a forward-backward heat equation

Summary. Galerkin and weighted Galerkin methods are proposed for the numerical solution of parabolic partial differential equations where the diffusion coefficient takes different signs. The approach is based on a simultaneous discretization of space and time variables by using continuous finite element methods. Under some simple assumptions, error estimates and some numerical results for both Galerkin and weighted Galerkin methods are presented. Comparisons with the previous methods show that new methods not only can be used to solve a wider class of equations but also require less regularity for the solution and need fewer computations. Received March 3, 1995     

KdV方程的延拓结构

利用外微分形式系统和Lie代数表示理论提出了求解非线性波方程Lax对的延拓结构理论,该方法是构造非线性波方程Lax对的系统最有效的方法.其关键在于如何给出延拓代数的具体表示,如微分算子表示或矩阵表示.如果一个非线性波方程具有非平凡的延拓代数,则称其延拓代数可积,本篇论文主要利用延拓结构理论,讨论KdV方程的解,同时给出...     

Numerical stability of the KdV equation with a negative forcing

The waves at the free surface waves of an incompressible and inviscid fluid in a two dimensional domain with horizontal rigid flat bottom with a small obstruction are considered. A time dependent KdV equation with a negative forcing is derived and studied both theoretically and numerically. The existence of a negative solitary-wave-like solution of the equation near the Froude number is proved and the numerical stability of the solution is also studied. The numerical stability of the positive both symmetric and unsymmetric solitary-wave-like solutions are also studied. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions of the KdV equation elliptic int

Two ansatzes of the Krichever curves for solutions of the KdV equation that are elliptic int are considered. Examples are given.State Academy of Aviation Instrument Manufacture, St. Petersburg. Translated from Teoreticheskaya i Mathematicheskaya Fizika, Vol. 100, No. 2, pp. 183–198, August, 1994.     

On the long time behaviour of a generalized KdV equation

We consider the Cauchy problem for the generalized Korteweg-de Vries equation% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabgkGi2oaaBaaaleaacaaIXaaabeaakiaadwhacqGHRaWkcqGH% ciITdaWgaaWcbaGaamiEaaqabaGccaGGOaGaeyOeI0IaeyOaIy7aa0% baaSqaaiaadIhaaeaacaaIYaaaaOGaaiykamaaCaaaleqabaGaeqyS% degaaOGaamyDaiabgUcaRiabgkGi2oaaBaaaleaacaWG4baabeaakm% aabmGabaWaaSaaaeaacaWG1bWaaWbaaSqabeaacqaH7oaBaaaakeaa% cqaH7oaBaaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa!56D5!\[\partial _1 u + \partial _x ( - \partial _x^2 )^\alpha u + \partial _x \left( {\frac{{u^\lambda }}{\lambda }} \right) = 0\]where is a positive real and and integer larger than 1. We obtain the detailed large distance behaviour of the fundamental solution of the linear problem and show that for 1/2 and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeU7aSjabg6da+iabeg7aHjabgUcaRmaalaaabaGaaG4maaqa% aiaaikdaaaGaey4kaSYaaeWaceaacqaHXoqydaahaaWcbeqaaiaaik% daaaGccqGHRaWkcaaIZaGaeqySdeMaey4kaSYaaSaaaeaacaaI1aaa% baGaaGinaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaigdacaGGVa% GaaGOmaaaaaaa!4FF7!\[\lambda > \alpha + \frac{3}{2} + \left( {\alpha ^2 + 3\alpha + \frac{5}{4}} \right)^{1/2} \], solutions of the nonlinear equation with small initial conditions are smooth in the large and asymptotic when t± to solutions of the linear problem.     

Finite-gap elliptic solutions of the KdV equation

A method is proposed for constructing finite-gap elliptic inx or/and int solutions of the Korteweg-de Vries equation. Dynamics of poles for two-gap elliptic solutions of the KdV equation are considered. Numerous examples of new elliptic solutions of the KdV equation are given.Dedicated to the memory of J.-L. Verdier     

Traveling waves for a dissipative modified KdV equation

In this paper we consider a dispersive–dissipative nonlinear equation which can be regarded as a dissipation perturbed modified KdV equation, governing the evolution of long waves in an elastic rod immersed inside a viscoelastic medium. Using geometric singular perturbation theory, a construction of traveling waves for the equation is shown. This also is illustrated by presenting some numerical calculations.     

Galerkin method for a third-order differential-operator equation

We study the Galerkin method for a third-order differential-operator equation with self-adjoint leading operator A and subordinate linear operator K(t) in a separable Hilbert space. We prove a theorem on the existence and uniqueness of a strong solution of the original problem. We derive estimates for the accuracy of the approximate solutions constructed by the Galerkin method. An application of the suggested method to the solution of a model problem is described.