New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

特征值问题的应用简述

从矩阵的特征问题入手,引出常系数线性齐次微分方程求解的特征方程方法;利用分离变量法求解热传导方程,引入拉普拉斯方程的特征问题,给出求解过程,并给出热方程的解的渐近稳定性.     

Approximate Solution of the Kuramoto-Shivashinsky Equation on an Unbounded Domain

The main goal of this paper is to approximate the Kuramoto-Shivashinsky (K-S for short) equation on an unbounded domain near a change of bifurcation, where a band of dominant pattern is changing stability. This leads to a slow modulation of the dominant pattern. Here we consider PDEs with quadratic nonlinearities and derive rigorously the modulation equation, which is called the Ginzburg-Landau (G-L for short) equation, for the amplitudes of the dominating modes.     

The Self-similar Solution for Ginzburg-Landau Equation and Its Limit Behavior in Besov Spaces

In this paper, we study the limit behavior of self-similar solutions for the Complex Ginzburg-Landau (CGL) equation in the nonstandard function space E_{s,p}. We prove the uniform existence of the solutions for the CGL equation and its limit equation in E_{s,p}. Moreover we show that the self-similar solutions of CGL equation converge, globally in time, to those of its limit equation as the parameters tend to zero. Key Words Ginzburg-Landau equation; Schrödinger equation; self-similar solution; limit behavior.     

一般区间上柱Bessel方程本征值问题特征问题根的Matlab算法

Root of characteristic equation for cylindrical Bessel equation eigenvalue prob-lems on general interval is of great real physical importance at engineering and physical. First, the characteristic equation of cylindrical Bessel equation eigenvalue problem on general interval is given, second, by mean of compared method, we obtaining roots of characteristic equation with Matlab program is discussed.     

Automatic differentiation of explicit Runge-Kutta methods for optimal control

This paper considers the numerical solution of optimal control problems based on ODEs. We assume that an explicit Runge-Kutta method is applied to integrate the state equation in the context of a recursive discretization approach. To compute the gradient of the cost function, one may employ Automatic Differentiation (AD). This paper presents the integration schemes that are automatically generated when differentiating the discretization of the state equation using AD. We show that they can be seen as discretization methods for the sensitivity and adjoint differential equation of the underlying control problem. Furthermore, we prove that the convergence rate of the scheme automatically derived for the sensitivity equation coincides with the convergence rate of the integration scheme for the state equation. Under mild additional assumptions on the coefficients of the integration scheme for the state equation, we show a similar result for the scheme automatically derived for the adjoint equation. Numerical results illustrate the presented theoretical results.     

Travelling Wave Solutions for Generalized Fisher Equation

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ［１，２］，ＡｒｏｎｓｏｎａｎｄＷｅｉｎｂｅｒｇｅｒｈａｖｅｓｔｕｄｉｅｄｓｙｓｔｅｍａｔｉｃｌｙｔｈｅｓｃａｌａｒｎｏｎｌｉｎｅａｒｄｉｆｆｕ－ｓｉｏｎｅｑｕａｔｉｏｎｉｎｏｎｅｓｐａｃｅｖａｒｉａｂｌｅｕｔ＝ｕｘｘ＋φ（ｕ），（１．１）ｗｈｅｒｅｕ＝ｕ（ｘ，ｔ）ａｎｄφ（ｕ）ｉｓａｎｏｎｌｉｎｅａｒｆｕｎｃｔｉｏｎ．Ｅｑｕａｔｉｏｎ（１．１）ａｒｉｓｅｓｉｎｓｅｖｅｒａｌａｐｐｌｉｃａ－ｔｉｏｎｓ；Ｓｅｅ［１，２］ａｎｄ［３］ｆｏｒｉｎｆｏｒｍａｔｉｏｎａ…     

Application of rational expansion method for stochastic differential equations

By means of the Hermite transformation, a new general ansätz and the symbolic computation system Maple, we apply the Riccati equation rational expansion method [24] to uniformly construct a series of stochastic non-traveling wave solutions for stochastic differential equations. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions. The method can also be applied to solve other nonlinear stochastic differential equation or equations.     

Special Monodromy Groups and the Riemann-Hilbert Problem for the Riemann Equation

In this paper, we solve the Riemann-Hilbert problem for the Riemann equation and for the hypergeometric equation. We describe all possible representations of the monodromy of the Riemann equation. We show that if the monodromy of the Riemann equation belongs to SL(2, ℂ), then it can be realized not only by the Riemann equation, but also by the more special Riemann-Sturm-Liouville equation. For the hypergeometric equation, we construct a criterion for its monodromy group to belong to SL(2, ℤ).__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 753–767.Original Russian Text Copyright ©2005 by V. A. Poberezhnyi.     

浅水波方程的初边值问题

本文用Kato关于拟线性演化方程的初值问题的存在性定理证明了浅水波方程在半无界直线上初值问题局部解的存在性。用解的先估计证明了整体解的存在性或解的Blow-up性质，并给出了解关于x的渐近估计。     

A study on KdV and Gardner equations with time-dependent coefficients and forcing terms

In this work we study the KdV equation and the Gardner equation with time-dependent coefficients and forcing term for each equation. A generalized wave transformation is used to convert each equation to a homogeneous equation. The soliton ansatz will be applied to the homogeneous equations to obtain soliton solutions.     

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.     

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984-1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.     

Representations of super-holomorphic functions and their applications

In this paper, we give three kinds of Almansi representations for super-polyharmonic functions. As applications of these representations, we investigate two kinds of boundary value problems in superspace and construct solutions to the super-Poisson equation and the inhomogeneous super-polyharmonic equation.     

A numerical approach for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation

In this article, we have introduced a Taylor collocation method, which is based on collocation method for solving initial-boundary value problem describing the process of cooling of a semi-infinite body by radiation. This method is based on first taking the truncated Taylor expansions of the solution function in the fractional differential equation and then substituting their matrix forms into the equation. Using collocation points, we have the system of nonlinear algebraic equation. Then, we solve the system of nonlinear algebraic equation using Maple 13 and we have the coefficients of Taylor expansion. In addition, numerical results are presented to demonstrate the effectiveness of the proposed method.     

Analysis of a Diagonal Form of the Fast Multipole Algorithm for Scattering Theory

Discretisation of the integral equations of acoustic scattering yields a system of linear equations with full coefficient matrices. In recent years a number of fast algorithms for the solution of this system have been proposed. In this paper we present a complete analysis for a fast multipole method for the Helmholtz equation. A one-level diagonal form of the multipole method is applied to a hypersingular integral equation arising from 2d scattering theory. The error of the approximation is analysed and the results used to establish the complexity of the method.     

Remarks on a semilinear heat equation with integral boundary conditions

For a semilinear heat equation we consider a nonlocal boundary problem. On the basis of the solution of a Dirichlet problem for a parabolic equation and Volterra integral equation we establish the well-posedness for the nonlocal problem, which generalizes some recent results.     

KdV方程和Kupersmidt方程的递推求解公式

根据Ｐａｉｎｌｅｖｅ分析的已知结果,给出了ＫｄＶ方程和Ｋｕｐｅｒｓｃｈｍｉｄｔ方程的新的递推求解公式。     

Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

Small-divisor equation of higher order with large variable coefficient and application to the coupled KdV equation

In this paper, we establish an estimate for the solutions of small-divisor equation of higher order with large variable coefficient. Then by formulating an infinite-dimensional KAM theorem which allows for multiple normal frequencies and unbounded perturbations, we prove that there are many periodic solutions for the coupled KdV equation subject to small Hamiltonian perturbations.