Solving mKdV-sinh-Gordon equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ODE method is developed for solving the mKdV-sinh-Gordon equation. As a result, many explicit and exact solutions including some new formal solutions are successfully picked up for the mKdV-sinh-Gordon equation by this approach.

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 〈 α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.

Solving (2+1)-dimensional sine-Poisson equation by a modified variable separated ordinary differential equation method

By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ordinary differential equation method is presented for solving the （2 ＋ 1）-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the （2 ＋ 1）-dimensional sine-Poisson equation are derived in a simple manner by this technique.

Some discussions about method for solving the variable separating nonlinear models

Through analysing the exact solution of some nonlinear models, the role of the variable separating method in solving nonlinear equations is discussed. We find that rich solution structures of some special fields of these equations come from the nonzero seed solution. However, these nonzero seed solutions is likely to result in the divergent phenomena for the other field component of the same equation. The convergence and the signification of all field components should be discussed when someone solves the nonlinear equation using the variable separating method.

On the feasibility of variable separation method based on Hamiltonian system for plane magnetoelectroelastic solids

The eigenvalue problem of an infinite-dimensional Hamiltonian operator appearing in the isotropic plane magnetoelectroelastic solids is studied. First, all the eigenvalues and their eigenfunctions in a rectangular domain are solved directly. Then the completeness of the eigenfunction system is proved, which offers a theoretic guarantee of the feasibility of variable separation method based on a Hamiltonian system for isotropic plane magnetoelectroelastic solids. Finally, the general solution for the equation in the rectangular domain is obtained by using the symplectic Fourier expansion method.

Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations

Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.

Variable Separation and Exact Solutions for the Kadomtsev-Petviashvili Equation

In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing the method of variable separation. Abundant exact solutions including global smooth solutions and local blow up solutions are obtained. These solutions would contribute to studying the behavior and blow up properties of the solution of the Kadomtsev-Petviashvili Equation.

A SPECIAL METHOD OF FOURIER SERIES WHICH IS EQUAL TO THE METHOD OF SEPARATION OF VARIABLES ON BOUNDARY VALUE PROBLEM

By the separation of singularity, a special Fourier series solution of the boundary value problem for plane is obtained, which can satisfy all boundary conditions and converges rapidly. II is proved that the solution is equal to the result of separation of variables. As a result, the non-linear characteristic equations resulting from the method of separation of variables are transformed into polynomial equations that can provide a foundation for approximate computation and asymptotic analysis.

New Families of Rational Form Variable Separation Solutions to(2+1)-Dimensional Dispersive Long Wave Equations

With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.

A New 2 + 1-Dimensional Integrable Variable Coefficient Toda Equation

In this paper, a new integrable variable coefficient Toda equation is proposed by utilizing a generalized version of the dressing method. At the same time, we derive the Lax pair of the new integrable variable coefficient Toda equation. The compatibility condition is given, which insures that the new Toda equation is integrable. To further analyze the character of the Toda equation, we derive one soliton solution of the obtained Toda equation by using separation of variables.

Prediction and Analysis of the Force and Shape Parameters in Variable Gauge Rolling

Variable gauge rolling is a new process to obtain a plate for which the thickness changes continuously by continuously and dynamically adjusting the roll gap upward and downward in the rolling process.This technology is an efective method for producing lightweight,low-cost,and economical plates.However,variable gauge rolling is an unsteady process,and the changes in the force and deformation parameters are complex.In this research,based on the minimum energy theory of the variational principle and considering the characteristics of the roll movement and workpiece deformation comprehensively,the internal plastic deformation,friction,shear and tension powers,and the minimum result of the total power functional in upward and downward rolling are obtained with the frst integral and then with a variation of adopting the specifc plastic power and strain rate vector inner product.The analytical results of the deformation and force parameters are also established using the variational method.Then the precision of this model is certifed using the measured values in a medium plate hot rolling plant and the experimental data for Tailor Rolled Blank rolling.Good agreement is found.Additionally,the variation rule of bite angle,neutral angle,and location neutral points are shown,and the change mechanism of the friction parameter on the stress state efect coefcient is given in variable gauge rolling.This research proposes a new mathematical model for rolling process control that provides a scientifc basis and technical support for obtaining an accurate section shape in variable gauge rolling production.

Functional Separable Solutions to Nonlinear Diffusion Equations by Group Foliation Method

We consider the functional separation of variables to the nonlinear diffusion equation with source and convection term： ut = （A（x）D（u）ux）x ＋ B（x）Q（u）, Ax ≠ 0. The functional separation of variables to this equation is studied by using the group foliation method. A classification is carried out for the equations which admit the function separable solutions. As a consequence, some solutions to the resulting equations are obtained.

拉索-阻尼器系统的改进实模态分析方法

阻尼减振是拉索振动控制的主要措施,目前拉索-阻尼器系统的主要分析方法为复模态法.虽然采用复模态法可以精确分析拉索-阻尼器系统的阻尼比和频率等主要特征参数,但采用复模态法求出的拉索复振型由于不包含减振所需要的参数而被研究较少,而系统的振型对研究阻尼器的减振效果又具有非常重要的意义.文中首先采用复模态法求解了拉索-阻尼器系统,基于复振型的物理意义,引入相位函数,提出了一种改进的实数分离变量法,分析了阻尼力在拉索中的传递特征;建立了拉索-阻尼器系统的改进实模态分析方法,采用该方法求解得到的非线性运动方程通解与复模态法一致.基于该新方法计算得到的实模态振型,分析了阻尼器对拉索实际振型以及相位的影响.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation： ut = （A（x）D（u）ux^n）x＋ B（x）Q（u）, Ax≠0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

Improved delayed detached-eddy simulation of massive separation around triple cylinders

The massively separated flow past triple cylin- ders （TriC） in tandem arrangement is simulated using the improved delayed detached-eddy simulation （IDDES） method based on the shear stress transport （SST） model, coupled with the high order adaptive dissipation scheme. The spacing between adjacent cylinders is sub-critical （1.435D）. IDDES prediction of two cylinders （TC） with the same spacing is compared to experimental data for validation, and the numerical results agree well with the available measurements, except for the asymmetry in the gap region. The flow past TriC is investigated using the same method. Generally, the mean flow quantities past TriC, such as the velocity, pressure, and vorticity, are similar to the corresponding components of TC. However, the pressure fluctuations on the TriC surface are uniformly larger than those on TC. Meanwhile, the instantaneous flows past TriC are much more complex. The periodical blockage in the first gap region is found in the TriC case and leads to the up-and-down movement of shear layer in the second gap region.

基于Wilson-θ显式算法的动载荷快速定位及重建方法

通过将Wilson-θ数值算法转变为一种等效的显式表达式,由此建立了一种基于Wilson-θ显式算法的动载荷识别方法。在选取合适的参数θ值以后,该算法是无条件稳定的,并且避免了以往隐式识别算法中由于递推计算产生误差累积导致识别结果严重发散的问题。通过引入响应系数矩阵,该显式算法可以灵活地选取响应类型用于载荷识别计算。同时结合分离变量方法,将载荷位置信息从建立的模型矩阵中提取出来,由此减少矩阵求逆的次数,提高载荷定位计算的效率。结合简支梁的仿真算例与试验测试对该载荷识别方法进行验证。研究结果表明,基于Wilson-θ显式算法的动载荷快速定位及重建方法能够有效地识别得到载荷位置及相应的时间历程。相对于传统载荷识别方法,该方法计算结果的准确性更好,且运算效率更高。

聚焦Kundu-Eckhaus方程中畸形波的奇异动力学行为研究

该文基于Darboux变换的相关结果,利用变量分离法推导出了聚焦Kundu-Eckhaus方程的一般高阶畸形波解.然后通过一些图形详细地讨论了这些畸形波解的动力学行为.特别地,这里可以观察到一个四瓣形畸形波和三眼形畸形波是可以共存的,这是完全不同于之前存在的四眼形畸形波.研究结果表明,该文所研究的畸形波结构比在著名的非线性Schrödinger方程中的畸形波结构更为丰富.

隧道拱顶渗漏稳态渗流场的解析研究

针对隧道运营中发生拱顶渗漏且无泥沙涌入的稳态渗流情况,结合保角变换法和分离变量法推导出隧道拱顶渗漏稳态渗流场的显式解析解.应用PLAXIS有限元软件建立数值模型,从渗流场分布、衬砌水压力两方面验证所得解的正确性,通过与现有解析解、数值解、试验结果的渗流量对比来验证所得解的准确性.进行参数分析,探究隧道埋深、渗漏宽度对衬砌水压力和渗流量的影响规律.结果表明,衬砌水压力在以渗漏位置为中心±60°范围内明显降低;随着隧道埋深的增大,衬砌最大水压力减小,渗流量先减小后增大,且渗流量最小值对应的隧道埋深随着地表水头的增大而增大;随着渗漏宽度的增大,衬砌最大水压力减小,渗流量增大,当渗漏宽度大于0.05 m时,渗漏宽度变化对水压力和渗流量的影响较小;在渗漏宽度较小时进行渗漏控制的效果明显.

Exactly solving some typical Riemann-Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL)fractional equations is very difficult.We propose a general method of separation of variables to study the problem.We obtain several general results and,as applications,we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation.In particular,we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation.In addition,we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.

一类特殊形式的时间分数阶Navier-Stokes方程的解

本文考虑了一类特殊形式的时间分数阶Navier-Stokes方程的解,采用分离变量法对方程进行变量分离,得到仅关于空间变量和仅关于时间变量的两个方程,前者是一奇异的Sturm-Liouville问题,利用Bessel函数求解。后者则是一个关于时间的分数阶常微分方程,分别采用积分变换、算子方法和Adomian分解法对其求解,得到的解一致。