求强非线性系统次谐共振解的MLP方法

定义了一个新的参数变换α=α(ε,nω₀/m,ω₁),扩展了改进的LP方法的应用范围,使该方法能够求强非线性系统的次谐共振解.研究了Duffing方程的1/3亚谐和3次超谐共振解以及Vander Pol-Mathieu方程1/2亚谐共振解,这些例子说明近似解和数值解相当吻合.     

图的非负广义邻接矩阵的谱半径的一些界（英文）

顶点数为n,边数为m的简单图G的非负广义邻接矩阵定义为U(G)=γAA(G)+γ_II(G)+γ_JJ(G)+γ_DD(G),其中γ_A,γ_I,γ_J,γ_D是一些非负实数,A(G)是图G的邻接矩阵,D(G)=diag(d₁,d₂,…,d_n),I(G)是单位矩阵,J(G)是全1矩阵.本文得到了谱半径ρ_U(G)的一些界,并刻画了达到这些界时的极图.此外还得到了ρ_Aα(G)的新界以及ρ_A(G),ρ_L(G)和ρ_Q(G)的已知界.     

一般Brown桥的解析解、数值解及仿真

对任意区间[t₀,t₁]任意初值为x₀和终值为x₁的Brown桥,即随机微分方程dx=(x₁-x)/(t₁-t)dt+dB(t) x(t₀)=x₀,给出其解析解(包括随机积分解和Fourier级数解),及Milstein数值解并进行期望和方差分析,求得期望函数、期望±标准差曲线和95%置信区间边界曲线.并对不同Brown运动对应的不同轨线实现进行仿真模拟.对一个非真实的近似解加以辨别.     

关于热冲击弹性问题的自由能及变分原理

在讨论耦合热弹性问题的变分原理的一些著作中,以弹性应变e_ij和温度变化值θ为状态参数的自由能φ(e_ij,θ)为自由能的这一表达式只适用于|θ|<0(绝对参考温度)的情况.在热冲击弹性问题中,温度变化值θ很大,甚至可以大过T₀同时,材料常数(λ,μ,γ,c等)随θ而发生变化,不再保持为常数.就这种情况,本文导出自由能的表达式.(0.1)式则为其特殊情况.将自由能的这一表达式引入变分原理,其欧拉方程将成为非线性.为了线性化,将热冲击作用的时间过程划分为若干足够小的时间元△t_k(△t_k=t_k-t_k-1,k=1,2,…,n).在△t_k中,温度变化θ_k很小,材料常数由t_k-1瞬时的温度场T_k-1=T_{x1,x2,x3,tk-1}确定,自由能φ_k可近似地采用(0.1)式的形式,从而得到变分原理的分段近似表达.     

不用Kirchhoff-Love假设的三维弹性板二级近似理论及其边界条件

前文[1]给出了不用Kirchhoff-Love假设的三维弹性板的一级近似理论及其边界条件。这个理论有6个微分方程求解6个待定平面函数,即u₀,u_α,A₍₀₎,S_(2)α,其中有3个方程为一组求解3个待定平面函数u₀,S_(2)α,而另一组3个方程求解另外3个待定平面函数u_α,A₍₀₎.它们的边界条件和这些方程一样,可以从本问题的广义变分原理的泛函变分的驻值条件求得,当板厚h和板宽α之比h/α很小时,这种解接近于经典薄板解,当h/α值较大时(如h/α≈0.3),这种解和经典薄板解,就有较大差别。但这种差别在h/α值的什么范围内是合理的这一问题,并不清楚,为了解决这一问题,我们必须研究本问题的二级近似理论。本文是前文的继续,我们将用本问题的广义变分原理的泛函变分驻值条件,导出9个微分方程和有关边界条件,用以求解9个二级近似解的待定平面函数u₀,u_α,A₍₀₎,A₍₁₎,S_(2)α,S_(3)α,把二级近似理论解和一级近似理论以及经典理论的解相比较,就能明确一级近似理论的适用范围,这里必须指出,二级近似理论也可以分成两组方程求解,求解过程也并不过分复杂。有关符号和前文相同,这里必须指出,二级近似理论也可以分成两组方程求解,求解过程也并不过分复杂.有关符号和前文相同,这里将不再重复。     

关于高超声速绕流问题的激波层方法

切尔诺依(ГГ.Черный)的高超声速绕流激波层的级数解法,不适用于(γ-1)/(γ+1)<<2/(γ十1)M2sin2β(γ=c_p/c_v为绝热指数,M为马赫数,β为激波倾角)的情形。本文只假定在激波邻近存在激波层,在此薄层内,气体密度很大,但不排除(γ-1)/(γ+1)<<2/(γ+1)M2sin2β的情形。     

广义Pochhammer-Chree方程的显式精确孤波解

首先对广义Pochhammer-Chre方程(PC方程)u_tt-u_ttxx+ru_xxt-(a₁u+a₂u2+a₃u3)_xx=0(r≠0)(Ⅰ)的孤波解u(ξ)建立了公式∫_-∞+∞[u'(ξ)]2dξ=1/12rv(C₊-C_-)3[3a₃(C₊+C_-)+2a₂]。由此推知:广义PC方程(Ⅰ)不可能有钟状孤波解,只可能有扭状孤波解;而广义PC方程u_tt-u_ttxx-(a₁u+a₂u2+a₃u3)_xx=0(Ⅱ)可能既有钟状孤波解又有渐近值满足3a₃(C₊+C_-)+2a₂=0的扭状孤波解。进一步求出了广义PC方程(Ⅰ)的扭状孤波解,求出了广义PC方程(Ⅱ)的钟状孤波解和渐近值满足2a₃(C₊+C_-)+2a₂=0的扭状孤波解。最后给出了广义PC方程u_tt-u_ttxx-(a₁u+a₃u3+a₅u5)_xx=0(Ⅲ)的显式孤波解。     

二阶奇异非线性微分方程边值问题的正解

分别在0≤f₀+<M₁,m₁<f_∞-≤∞和0≤f_∞+<M₁,m₁<f₀-≤∞的情形下研究了非线性奇异边值问题u″+g(t)f(u)=0,0<t<1,αu(0)-βu′(0)=0,γu(1)+δu′(1)=0正解的存在性,其中f₀+=₀f(u)/u,f_∞-=_∞f(u)/u,f₀-=₀f(u)/u,f_∞+=_∞f(u)/u,g在区间[0,1]的端点可以具有奇性。     

自守L-函数集合的零点密度的整体上界估计

本文通过L-函数的整体积分幂矩,来推导某些自守L-函数集合的整体零点密度的上界估计.具体而言,假设I是某些自守表示π构成的集合,对任意π有非负系数c(π)且级数∑_π∈I c(π)收敛.假设■其中l≥1,0 <α≤1,θ≥α.则可以得到整体零点密度■的上界估计,这里N_π(σ,T₁,T₂)表示满足σ<β<1及T₁≤γ≤T₂的L(s,π)的零点ρ=β+iγ的个数.     

第一类不适定算子方程的形式解

本文在Sobolev空间W1/2中讨论第一类算子方程Au=f当A~(-1)不连续时的不适定求解问题。文中利用W1/2空间的再生核R_x(y)引入方程Au=f的形式解的概念,证明了在某些有界性条件下,形式解就是经典解;而且可由形式解的部分和直接得到稳定的数值解,这样就省去了目前常用的正则法与拟解法为得到数值解而求解泛函极值的过程。     

New periodic wave solutions for the shallow water equations and the generalized Klein–Gordon equation

The periodic wave solutions and the corresponding solitary solutions for the shallow water equations and the generalized Klein–Gordon equation are obtained by means of mapping method. The solutions obtained in this paper include as well the shock wave solution, complex line period, complex line soliton and rational solutions. Moreover, the obtained solutions are degenerated in terms of hyperbolic function solutions and trigonometric function solutions when the modulus m of the Jacobi elliptic function is driven to 1 and 0, respectively. The previously known periodic and solitary wave solutions are recovered. Many new results are presented.     

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.     

Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation

In this paper, the F-expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation.     

Variable-coefficient Jacobi elliptic function expansion method for (2+1)-dimensional Nizhnik-Novikov-Vesselov equations

In this paper, a variable-coefficient Jacobi elliptic function expansion method is proposed to seek more general exact solutions of nonlinear partial differential equations. Being concise and straightforward, this method is applied to the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations. As a result, many new and more general exact non-travelling wave and coefficient function solutions are obtained including Jacobi elliptic function solutions, soliton-like solutions and trigonometric function solutions. To give more physical insights to the obtained solutions, we present graphically their representative structures by setting the arbitrary functions in the solutions as specific functions.     

The generalizing Riccati equation mapping method in non-linear evolution equation: application to (2 + 1)-dimensional Boiti–Leon–Pempinelle equation

The tanh method is used to find travelling wave solutions to various wave equations. In this paper, the extended tanh function method is further improved by the generalizing Riccati equation mapping method and picking up its new solutions. In order to test the validity of this approach, the (2 + 1)-dimensional Boiti–Leon–Pempinelle equation is considered. As a result, the abundant new non-travelling wave solutions are obtained.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

（2＋1）维KP方程的三类精确解

研究了（2＋1）维KP方程的孤子解问题．应用Riccati方程映射法,得到了（2＋1）维KP方程的新的显式精确解的结构．根据得到的精确解结构,构造出了该方程的三类精确解．     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

New exact Jacobi elliptic functions solutions for the generalized coupled Hirota-Satsuma KdV system

In this paper, an generalized Jacobi elliptic functions expansion method with computerized symbolic computation is used for constructing more new exact Jacobi elliptic functions solutions of the generalized coupled Hirota-Satsuma KdV system. As a result, eight families of new doubly periodic solutions are obtained by using this method, some of these solutions are degenerated to solitary wave solutions and triangular functions solutions in the limit cases when the modulus of the Jacobi elliptic functions m → 1 or 0, which shows that the applied method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear partial differential equations arising in mathematical physics.     

Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev-Petviashvili equation

In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.