Explicit traveling wave solutions of five kinds of nonlinear evolution equations

First of all, by using Bernoulli equations, we develop some technical lemmas. Then, we establish the explicit traveling wave solutions of five kinds of nonlinear evolution equations: nonlinear convection diffusion equations (including Burgers equations), nonlinear dispersive wave equations (including Korteweg-de Vries equations), nonlinear dissipative dispersive wave equations (including Ginzburg-Landau equation, Korteweg-de Vries-Burgers equation and Benjamin-Bona-Mahony-Burgers equation), nonlinear hyperbolic equations (including Sine-Gordon equation) and nonlinear reaction diffusion equations (including Belousov-Zhabotinskii system of reaction diffusion equations).     

Group properties of the extended Green–Naghdi equations

The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

Linearized stability analysis of two-dimension Burnett equations

The linearized stability analyses of two-dimension Burnett equations were studied in present paper for the first time. The characteristic stability equation of two-dimension original Burnett equation was first derived and the characteristic curve was achieved. The material derivatives in original Burnett equations are then replaced with the Euler and Navier-Stokes equations. The stabilities of these two alternative Burnett equations are then analyzed. The linearized stability analyses show that the two-dimension original Burnett and Euler-based Burnett equations are not stable while the Navier-Stokes-based Burnett equations are stable. The critical Knudsen number for the original Burnett and Euler-based Burnett equations are 0.074 and 0.353, respectively. These critical Knudsen number are smaller than those of corresponding one-dimension equations. The two-dimension Burnett equations are more unstable than one-dimension equations.     

抽象函数空间的连续小波变换和微分方程

主要讨论了抽象函数的某些微分方程和相应的积分方程之间的关系;通过连续小波变换将这些微分方程能够转换为相应的积分方程;这些微分方程和相应的积分方程在弱收敛意义下是等价的.     

Integrating factors, adjoint equations and Lagrangians

Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.     

More common errors in finding exact solutions of nonlinear differential equations: Part I

In the recent paper by Kudryashov [11] seven common errors in finding exact solutions of nonlinear differential equations were listed and discussed in detail. We indicate two more common errors concerning the similarity (equivalence with respect to point transformations) and linearizability of differential equations and then discuss the first of them. Classes of generalized KdV and mKdV equations with variable coefficients are used in order to clarify our conclusions. We investigate admissible point transformations in classes of generalized KdV equations, obtain the necessary and sufficient conditions of similarity of such equations to the standard KdV and mKdV equations and carried out the exhaustive group classification of a class of variable-coefficient KdV equations. Then a number of recent papers on such equations are commented using the above results. It is shown that exact solutions were constructed in these papers only for equations which are reduced by point transformations to the standard KdV and mKdV equations. Therefore, exact solutions of such equations can be obtained from known solutions of the standard KdV and mKdV equations in an easier way than by direct solving. The same statement is true for other equations which are equivalent to well-known equations with respect to point transformations.     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

Explicit Traveling Wave Solutions to Nonlinear Evolution Equations

First of all, some technical tools are developed. Then the author studies explicit traveling wave solutions to nonlinear dispersive wave equations, nonlinear dissipative dispersive wave equations, nonlinear convection equations, nonlinear reaction diffusion equations and nonlinear hyperbolic equations, respectively.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

An examination of the homentropic Euler equations with averaged characteristics

This paper examines the properties of the homentropic Euler equations when the characteristics of the equations have been spatially averaged. The new equations are referred to as the characteristically averaged homentropic Euler (CAHE) equations. An existence and uniqueness proof for the modified equations is given. The speed of shocks for the CAHE equations are determined. The Riemann problem is examined and a general form of the solutions is presented. Finally, numerically simulations on the homentropic Euler and CAHE equations are conducted and the behaviors of the two sets of equations are compared.     

An iterative method for solving nonlinear functional equations

An iterative method for solving nonlinear functional equations, viz. nonlinear Volterra integral equations, algebraic equations and systems of ordinary differential equation, nonlinear algebraic equations and fractional differential equations has been discussed.     

层状二维流动的基本方程式

在很多海洋、大气等二维流动问题中所用的动力学方程往往沿用推广后的河流水力学方程或"纳维-斯托克斯方程"其中把湍流阻力项写成这样的方程式和湍流阻力项用到实际问题上去,无疑是存在着极大的局限性,而将导致矛盾百出.本文则从雷诺方程出发,把所有的物理量沿深度加以平均,求出平均以后的物理量所满足的运动方程,连续方程和扩散方程.     

一类非牛顿渗流方程爆破界的估计

本文利用拟线性常微分方程解的非存在性定理得到了一类拟线性反应扩散方程(非牛顿渗流方程)爆破界的估计,从而推广了半线性反应扩散方程(牛顿渗流方程)相应结果.     

Classifying Integrable Egoroff Hydrodynamic Chains

We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case. We find (2+1)-dimensional equations that are not just generalizations of the already known Khokhlov–Zabolotskaya and Boyer–Finley equations but are much more involved. These equations are parameterized by theta functions and by solutions of the Chazy equations. We obtain analogues of the dispersionless Hirota equations.     

Parabolic equations related to curve motion

We study curve motions based on differential equations. Curve motion equations are classified into two types: adaptive equations (which depend on the choice of coordinate systems) and non-adaptive equations. Examples from both types of equations are studied, and the global existence for these equations is proved based on integral estimates.     

Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

Second Order Scalar Functional Differential Equations with Piecewise Constant Arguments

The study of functional differential equations with piecewise constant arguments usually results in a study of certain related difference equations. In this paper we consider certain neutral functional differential equations of this type and the associated difference equations. We give conditions under which such equations with almost periodic time dependence will have unique almost periodic solutions, and for certain autonomous cases, we obtain certain stability results and also conditions for chaotic behavior of solutions. We are particularly concerned with such equations which are partially discretized versions of non-forced Duffing equations.     

Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.