耦合非线性波方程的指数吸引子

在文[1]的基础上，得到了耦合非线性波方程的指数吸引子的存在性．     

KD-拉回吸引的非自治动力系统拉回吸引子的存在性及其应用

利用拉回吸引子的存在性理论,证明了具有KD-拉回吸引的非自治动力系统拉回吸引子的存在性,拉回吸引子是单点集,是不变的.对无解域上的非自治反映扩散方程,证明了拉回指数吸引子的存在性,是方程唯一拉回指数吸引的稳定解.     

二维广义Ginzburg-Landau方程的指数吸引子

在文[1]的基础上，得到了二维广义的Ginzburg－Landau方程的指数吸引子的存在性．     

三维无界区域上弱阻尼非线性Schrodinger方程的指数吸引子

文［１］中证明了弱阻尼非线性Ｓｃｈｒｄｉｎｇｅｒ方程在无界区域ＲＮ（Ｎ≤３）上存在一个最大的紧吸引子．本文在此基础上得到了Ｒ３上指数吸引子的存在性     

推广的B-BBM方程的整体吸引子和指数吸引子

本文对耗散的推广的B－BBM方程的长时间动力学行为进行了研究，证明了该方程整体吸引子和指数吸引子的存在性。     

弱D-拉回指数吸引子的存在性

主要研究弱D-拉回指数吸引子的存在性.首先讨论了弱D-拉回指数吸引子与非紧性测度之间的关系,其次,建立了弱D-拉回指数吸引子存在性的一般方法,最后证明了外力项具有指数增长速度的反应扩散方程在H_0~1(Ω)中存在弱D-拉回指数吸引子.     

记忆型梁方程强解的全局吸引子

记忆型梁方程出现于—般的Kirchhoff粘弹性梁模型中．本文在记忆核满足指数衰退的条件下证明了系统的能量也是指数衰退的．进一步,通过对条件(C)的验证获得了系统强解的全局吸引子．     

吊桥方程指数吸引子的存在性

吊桥方程是工程数学中的一类重要数学模型.在非自治和自治两种情况下,吊桥方程紧整体吸引子的存在性已有许多结果.然而,这一问题指数吸引子的存在性还无人讨论.我们借助Eden等人提出的方法,证明了该问题指数吸引子的存在性.     

周期边界条件下磁流体动力学方程的指数吸引子

我们考虑周期边条件下二维磁流体动力学（MHD）方程，证明了指数吸引子的存在性并给出其分形维度的上界估计．     

有阻尼受迫sine-Gordon方程的全局周期吸引子

本文证明了当阻尼与扩散系数在一定的参数范围内时,有阻尼的受迫sineGordon方程的狄氏问题对于任意非自治时间周期受迫力均具有唯一的指数吸引有界集的周期解．并且,如果受迫力是自治的,则全局吸引子恰是系统唯一的指数吸引有界集的平衡解．     

变厚度扁锥壳的非线性固有频率

借助于变厚度圆薄板非线性动力学变分方程和协调方程,给出了变厚度扁薄锥壳的非线性动力学变分方程和协调方程·　假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取变厚度扁锥壳中心最大振幅为摄动参数,采用摄动变分法,将变分方程和微分方程线性化·　对周边固定的圆底变厚度扁锥壳的非线性固有频率进行了求解;一次近似得到了变厚度扁锥壳的线性固有频率,三次近似得到了变厚度扁锥壳的非线性固有频率,且绘出了固有频率与静载荷、最大振幅、变厚度参数的特征曲线图·　为动力工程提供了有价值的参考·     

Symmetry reduction,exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation

We show that the Benjamin–Bona–Mahoney (BBM) equation with power law nonlinearity can be transformed by a point transformation to the combined KdV–mKdV equation, that is also known as the Gardner equation. We then study the combined KdV–mKdV equation from the Lie group-theoretic point of view. The Lie point symmetry generators of the combined KdV–mKdV equation are derived. We obtain symmetry reduction and a number of exact group-invariant solutions for the underlying equation using the Lie point symmetries of the equation. The conserved densities are also calculated for the BBM equation with dual nonlinearity by using the multiplier approach. Finally, the conserved quantities are computed using the one-soliton solution.     

Lie group symmetries and Riemann function of Klein–Gordon–Fock equation with central symmetry

In the present paper Lie symmetry group method is applied to find new exact invariant solutions for Klein–Gordon–Fock equation with central symmetry. The found invariant solutions are important for testing finite-difference computational schemes of various boundary value problems of Klein–Gordon–Fock equation with central symmetry. The classical admitted symmetries of the equation are found. The infinitesimal symmetries of the equation are used to find the Riemann function constructively.     

COLE-HOPF QUOTIENT AND EXACT SOLUTIONS OF THE GENERALIZED FITZHUGH-NAGUMO EQUATIONS

1IntroductionTheFitzhugfh-Nagullloequatioll'at=AL.r.r '(L('u--a)(1--'u)(1.1jisallimportantllolllinearreactioll-diffusiollequation.Itisusedwidelyillcircuittheory,I)l()I()gy(llldoillerfields.Thisequationhasobviouslythreetrivialsolutiolls:it=0,a.1.Bysotzt,illgIc=f(x--ct),onecallobtaillitsthreeheteroclinicsohltions.Inordertofill'litslllor(}axal('t~sollltiolls.someInethodswhicllareoillyllseftllforilltegrablesystellis11ave})chillgclleri\lizc(\Itotillsuoniutegrableequation.K;twaharaandTal,aka[1…     

Solving optimal control problems for the unsteady Burgers equation in COMSOL Multiphysics

The optimal control of unsteady Burgers equation without constraints and with control constraints are solved using the high-level modelling and simulation package COMSOL Multiphysics. Using the first-order optimality conditions, projection and semi-smooth Newton methods are applied for solving the optimality system. The optimality system is solved numerically using the classical iterative approach by integrating the state equation forward in time and the adjoint equation backward in time using the gradient method and considering the optimality system in the space-time cylinder as an elliptic equation and solving it adaptively. The equivalence of the optimality system to the elliptic partial differential equation (PDE) is shown by transforming the Burgers equation by the Cole-Hopf transformation to a linear diffusion type equation. Numerical results obtained with adaptive and nonadaptive elliptic solvers of COMSOL Multiphysics are presented both for the unconstrained and the control constrained case.     

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

A Multi-Symplectic Scheme for RLW Equation

The Hamiltonian and multi-symplectic formulations for RLW equation are considered in this paper. A new twelve-point difference scheme which is equivalent to multi-symplectic Preissmann integrator is derived based on the multi-symplectic formulation of RLW equation. And the numerical experiments on solitary waves are also given. Comparing the numerical results for RLW equation with those for KdV equation, the inelastic behavior of RLW equation is shown.     

N-soliton solutions for the integrable bidirectional sixth-order Sawada-Kotera equation

In this work, the integrable bidirectional sixth-order Sawada-Kotera equation is examined. The equation considered is a KdV6 equation that was derived from the fifth order Sawada-Kotera equation. Multiple soliton solutions and multiple singular soliton solutions are formally derived for this equation. The Cole-Hopf transformation method combined with the Hirota’s bilinear method are used to determine the two sets of solutions, where each set has a distinct structure.     

Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.     

关于广义KPP方程的数值解

广义KPP(Kolmogorov-Petrovskii-Piskunov)方程是一个积分微分方程.为了要研究其数值解,我们首先将该方程转化为一个非线性双曲型方程,然后构造了一个线性化的差分格式,得到了差分格式解的存在唯一性,利用能量不等式证明了差分格式二阶收敛性和关于初值的无条件稳定性,数值结果验证了本文提出的方法.