The integral bifurcation method combined with factoring technique for investigating exact solutions and their dynamical properties of a generalized Gardner equation

It is well known that it is difficult to obtain exact solutions of some partial differential equations with highly nonlinear terms or high order terms because these kinds of equations are not integrable in usual conditions. In this paper, by using the integral bifurcation method and factoring technique, we studied a generalized Gardner equation which contains both highly nonlinear terms and high order terms, some exact traveling wave solutions such as non-smooth peakon solutions, smooth periodic solutions and hyperbolic function solutions to the considered equation are obtained. Moreover, we demonstrate the profiles of these exact traveling wave solutions and discuss their dynamic properties through numerical simulations.     

A reaction-diffusion equation and its traveling wave solutions

In the present paper, we study a non-linear reaction-diffusion equation, which can be considered as a generalized Fisher equation. An exact solution and traveling wave solutions to the generalized Fisher equation are obtained by means of the Cole-Hopf transformation and the Lie symmetry method.     

Exact solutions to an evolution equation of plastic layer flow on a plane

The flow of a thin plastic layer between two rigid plates approaching each other in the normal direction is considered. The kinematics of plastic layer flow is studied. An evolution equation describing the free boundary of the flow region is derived. The similarity solutions to this equation are analyzed. It is shown that the evolution equation can be reduced to a particular case of the nonlinear heat conduction equation. New exact particular solutions to the evolution equation are obtained using the variable separation method and the method of self-similar transformations.     

A generalized auxiliary equation method and its applications

In this paper, a generalized auxiliary equation method with the aid of the computer symbolic computation system Maple is proposed to construct more exact solutions of nonlinear evolution equations, namely, the higher-order nonlinear Schrödinger equation, the Whitham–Broer–Kaup system, and the generalized Zakharov equations. As a result, some new types of exact travelling wave solutions are obtained, including soliton-like solutions, trigonometric function solutions, exponential solutions, and rational solutions. The method is straightforward and concise, and its applications are promising.     

Symmetry analysis of modified 2D Burgers vortex equation for unsteady case

In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations(PDEs).     

WKB method for analyzing the vibrations of a Meshcherskii pendulum

TheWKB method is used to construct an approximate analytic solution of the equation of small nonstationary vibrations of the Meshcherskii mathematical pendulum. The versions of linear and nonlinear variation in the pendulum mass are considered. Calculations showed that, under certain restrictions on the pendulum parameters, the approximate solutions constructed in elementary functions are a good approximation to the exact results.     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

Shape analysis and damped oscillatory solutions for a class of nonlinear wave equation with quintic term简

This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.     

A NEW STATE SPACE SOLUTION FOR RECTANGULAR THICK LAMINATES WITH CLAMPED EDGES Hu Wenfeng Liu Yihua

通过把固支边上的边界位移函数作为状态变量引入状态方程,得到了含固支边矩形单层与叠层厚板的精确解. 在求解过程中,将非齐次状态方程的求解变为齐次状态方程的求解,省略了求解待定常数的中间过程,使求解过程变得简单. 所得到的解能够严格满足固支边界条件,在同一材料层内不需作分层处理,因而更加精确. 此外,对固支边的应力提出了新的计算方法,能够得到更精确的边界应力. 算例表明,本文解比现有精确解收敛快,与有限元解吻合的更好,尤其是在固支边处体现得更加明显.     

An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.     

Complete group classification and exact solutions to the extended short pulse equation

In this paper, the complete group classification is performed on the extended short pulse equation (ESPE), which including many important non-linear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary parameters of the equation. Furthermore, the symmetry reductions and exact solutions to the short pulse types of equations are investigated, and the physical significance of the solutions are considered from the transformation group point of view.     

Computation of flow of viscoelastic fluids by parameter differentiation

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three-dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first-order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.     

考虑液面波动和液体压缩时一侧受液作用悬臂梁的横振分析

本文分析一侧部分受有液体作用悬臂梁的横向自由振动，同时考虑了液面波动和液体可压缩性对悬臂梁自振特性的影响。利用一组广义三角级数的正交完备性，求得了悬臂梁与液体耦联振动的振型函数和频率方程的精确解析解，最后给出了几个数值算例。     

Auto-Darboux Transformation and Exact Solutions of the Brusselator Reaction Diffusion Model

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NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

Two exact solutions of the DPL non-Fourier heat conduction equation with special conditions

This paper presents two exact explicit solutions for the three dimensional dual-phase lag （DLP） heat conduction equation, during the derivation of which the method of trial and error and the authors＇ previous experiences are utilized. To the authors＇ knowledge, most solutions of 2D or 3D DPL models available in the literature are obtained by numerical methods, and there are few exact solutions up to now. The exact solutions in this paper can be used as benchmarks to validate numerical solutions and to develop numerical schemes, grid generation methods and so forth. In addition, they are of theoretical significance since they correspond to physically possible situations. The main goal of this paper is to obtain some possible exact explicit solutions of the dual-phase lag heat conduction equation as the benchmark solutions for computational heat transfer, rather than specific solutions for some given initial and boundary conditions. Therefore, the initial and boundary conditions are indeterminate before derivation and can be deduced from the solutions afterwards. Actually, all solutions given in this paper can be easily proven by substituting them into the governing equation.     

非圆截面杆中的非线性扭转波及其行波解

研究了非圆截面杆中非线性扭转波的传播特性.由于非圆截面杆的扭转运动会伴随有横截面的翘曲,这种翘曲运动将引起扭转波的弥散.如果同时考虑有限扭转变形和翘曲弥散的共同作用,将会得到非线性扭转波的方程.在相平面上,对非线性扭转波动方程进行定性分析,结果表明,在一定条件下方程存在同宿轨道或异宿轨道,分别相应于方程的孤波解或冲击波解.本文利用Jacobi椭圆函数展开法,对该非线性方程进行求解,得到了非线性波动方程的三类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.这些条件与定性分析的结果相一致.     

Convergence and exact solutions of spline finite strip method using unitary transformation approach

The spline finite strip method(PSM) is one of the most popular numerical methods for analyzing prismatic structures.Efficacy and convergence of the method have been demonstrated in previous studies by comparing only numerical results with analytical results of some benchmark problems.To date,no exact solutions of the method or its explicit forms of error terms have been derived to show its convergence analytically. As such,in this paper,the mathematical exact solutions of spline finite strips in the plat...     

A new method to the (2+1)-dimensional modified KP equation

By means of the auxiliary ordinary differential equation method, we have obtained many solitary wave solutions, periodic wave solutions and variable separation solutions for the (2+1)-dimensional KP equation. Using a mixed method, many exact solutions have been obtained.