Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.     

Exact solutions for the generalized Klein–Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method

In this paper, a suitable transformation and a so-called Exp-function method are used to obtain different types of exact solutions for the generalized Klein–Gordon equation. These exact solutions are in full agreement with the previous results obtained in Refs. [Sirendaoreji, Auxiliary equation method and new solutions of Klein–Gordon equations, Chaos, Solitons & Fractals 31 (4) (2007) 943–950; Huiqun Zhang, Extended Jacobi elliptic function expansion method and its applications, Communications in Nonlinear Science and Numerical Simulation, 12 (5) (2007) 627–635]. One of these exact solutions is compared with the approximate solutions obtained by the modified decomposition method. Accurate numerical results for a wider range of time are obtained after using different types of ADM-Padè approximation. Our results show that the Exp-function method is very effective in finding exact solutions for the problem considered while the modified decomposition method is very powerful in finding numerical solutions with good accuracy for nonlinear PDE without any need for a transformation or perturbation.     

Soliton solutions of a few nonlinear wave equations

This paper carries out the integration of a few nonlinear wave equations to obtain topological as well as non-topological soliton solutions. The mathematical techniques used to obtain the soliton solutions are He’s variational iteration method, the tanh method and the ansatz method. The nonlinear wave equations that are studied are coupled mKdV equations, Drinfeld-Sokolov equation and its generalized version. Finally, some numerical simulations are given to support the analytical solutions.     

SOLITON SOLUTIONS TO AN EQUATION IN 2+1 DIMENSIONS

This paper considers a 2+1 dimensional equation. A bilinear form for the equation and its 3-soliton solutions are obtained by the Hirota method. The N-soliton solution is given in the form of pfaffian. At the same time the proof of the solutions are given.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

一类非线性演化方程新的行波解

利用双函数法和吴消元法,得到了一类非线性演化方程在不同情况下的一系列显示精确解.Sinh-Gordon方程及Klein-Gordon方程作为该方程的特例也得到了相应的行波解.     

Multiple soliton solutions for the Bogoyavlenskii’s generalized breaking soliton equations and its extension form

In this work, two generalized breaking soliton equations, namely, the Bogoyavlenskii’s breaking soliton equation and its extended form, are examined. The complete integrability of these equation are justified. Multiple soliton solutions and multiple singular soliton solutions are formally derived for each equation. The additional terms of these equations do not kill the integrability of the typical breaking soliton equation. The Cole-Hopf transformation method and the simplified Hereman’s method are applied to conduct this analysis.     

The variational iteration method combined with improved generalized tanh-coth method applied to Sawada-Kotera equation

We obtain new exact solutions to generalized Sawada-Kotera equation. Using the variational iteration method combined with the improved generalized tanh-coth method, we construct new traveling wave solutions for the standard Sawada-Kotera equation and, by means of scaling, we obtain new solutions to general Sawada-Kotera equation. Periodic and soliton solutions are formally derived for both models.     

一类非线性Klein-Gordon方程的数值解

本文主要研究了一类非线性Klein-Gordon方程.利用Fourier谱方法对一类非线性Klein-Gordon方程的求解,给出了求解的离散过程,并通过了数值模拟与文献结果进行了对比.结果表明这种方法对于求解此类非线性Klein-Gordon方程具有很好的效果.     

The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

New exact travelling wave solutions of two nonlinear physical models

In this paper, an improved tanh function method is used with a computerized symbolic computation for constructing new exact travelling wave solutions on two nonlinear physical models namely, the quantum Zakharov equations and the (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions.The exact solutions are obtained which include new soliton-like solutions, trigonometric function solutions and rational solutions. The method is straightforward and concise, and its applications are promising.     

一类Burgers-KPP方程新的显式精确行波解

In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.     

非线性发展方程新的显式精确解

借助Ｍａｔｈｅｍａｔｉｃａ系统,采用三角函数法和吴文俊消元法,本文获得了著名的２＋１维ＫＰ方程的若干精确解,其中包括新的精确解和孤波解．在此基础上,进而得到著名ＫｄＶ方程、Ｈｉｒｏｔａ－Ｓａｔｓｕｍａ方程和耦合ＫｄＶ方程的一些精确解．     

一个包含伪Smarandache函数及其对偶函数的方程

研究一个包含伪Smarandache函数及其对偶函数方程的可解性,利用初等及组合方法给出了该方程的一系列正整数解,并证明了该方程的所有奇数解必为奇素数p（≥5）的方幂.     

The Exact Solutions for the Benjamin-Bona-Mahony Equation

The Benjamin-Bona-Mahony (BBM) equation represents the unidirectional propagation of nonlinear dispersive long waves, which has a clear physical background, and is a more suitable mathematical and physical equation than the KdV equation. Therefore, the research on the BBM equation is very important. In this article, we put forward an effective algorithm, the modified hyperbolic function expanding method, to build the solutions of the BBM equation. We, by utilizing the modified hyperbolic function expanding method, obtain the traveling wave solutions of the BBM equation. When the parameters are taken as special values, the solitary waves are also derived from the traveling waves. The traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The modified hyperbolic function expanding method is direct, concise, elementary and effective, and can be used for many other nonlinear partial differential equations.     

Exact solutions for nonlinear partial differential equations via Exp‐function method

In this article, the Exp‐function method is applied to nonlinear Burgers equation and special fifth‐order partial differential equation. Using this method, we obtain exact solutions for these equations. The method is straightforward and concise, and its applications are promising. This method can be used as an alternative to obtain analytical and approximate solutions of different types of nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

三类conformable型分数阶偏微分方程的精确解

对于conformable型分数阶的Airy方程和Telegraph方程,利用泛函分离变量法和广义分离变量法求解了它们的精确解.对于无黏的conformable型分数阶Burgers方程,利用广义分离变量法求解了它的精确解.事实证明,分离变量法是一种简洁直接的求解方法.此外,还借助Maple软件绘制了一些解的三维图像.     

Auxiliary equation method for the mKdV equation with variable coefficients

By using solutions of an ordinary differential equation, an auxiliary equation method is described to seek exact solutions of nonlinear evolution equations with variable coefficients. Being concise and straightforward, this method is applied to the mKdV equation with variable coefficients. As a result, new explicit solutions including solitary wave solutions and trigonometric function solutions are obtained with the aid of symbolic computation.     

A note on the modified simple equation method applied to Sharma-Tasso-Olver equation

Jawad et al. have applied the modified simple equation method to find the exact solutions of the nonlinear Fitzhugh-Naguma equation and the nonlinear Sharma-Tasso-Olver equation. The analysis of the Sharma-Tasso-Olver equation obtained by Jawad et al. is based on variant of the modified simple equation method. In this paper, we provide its direct application and obtain new 1- soliton solutions.     

Error analysis of approximate solutions of non-linear Burgers’ equation

This paper deals with application of the maximum principle for differential equations to the finite difference method for determining upper and lower approximate solutions of the non-linear Burgers’ equation and their error range. In term of mathematical architecture, the paper is based on the maximum principle for parabolic differential equations to establish monotonic residual relations of the Burgers’ equation; and in terms of numerical method, it applies the finite difference method to discretize the equation, followed by use of the proposed Residual Correction Method for obtaining its optimal solutions under constraint conditions for inequalities. Derived by using this approach, the upper and lower transient approximate solutions are not just useful in analyzing the range of the maximum possible error between them and the analytic solutions correctly, and the numerical validations also indicate good accuracy in mean values of the upper and lower approximate solutions.