Fokas-Lenells方程的代数几何解

从一个给定的谱问题出发,利用Lenard梯度序列推导出Fokas-Lenells方程.随后,这个方程被分解为可解的常微分方程.基于Lax矩阵的有限阶展开,引入了椭圆坐标,从而,流可以在Abel-Jacobi坐标下被拉直.最后,利用Riemann θ函数得到了Fokas- Lenells方程的代数几何解的表示.     

一个2+1维可积方程的代数几何解

该文从1+1维的孤子方程出发,构造出一个2+1维在Lax意义下可积的方程.接着这个2+1维可积方程被分解为可解的常微分方程.随后引入超椭圆Riemann曲面和Abel-Jacobi坐标把流进行了拉直.再利用Riemannθ函数给出了这个2+1维方程的代数几何解.     

关于隐式代数曲面的一种几何连续性定义

本利用函数的幂级数给出了隐式代数曲面的一种几何连续性定义，证明了在此定义下的k阶几何连续即为k阶变尺度(rescaling)连续，并举例说明了其应用的方便性。     

几何流方程的分离变量解

利用不变子空间方法及拟设法,在变量变换作用下给出双曲几何流和Ricci流的各种分离变量解,包括乘法分离变量解和广义泛函分离变量解,并给出了这些解的性质分析.     

锥壳的位移解及应用*

本文提出求锥壳方程通解的另一种方法——位移法.文中根据文献[1]给出的壳体基本关系,导出锥壳一般弯曲问题的位移方程组,然后通过引入一个位移函数U(s,θ)(在极限情况下,就变为对于圆柱壳所引入的位移函数),从而将锥壳基本方程组化成关于位移函数U(s,θ)的8阶可解偏微分方程(控制微分方程).对于一般弯曲问题,该方程的一般解以广义超几何函数给出;对于轴对称弯曲问题,用Bessel函数给出其一般解.作为锥壳位移解法的应用,讨论了Winkler地基模式上的锥壳的轴对称弯曲问题,给出数值结果.     

构造方程模型解几何中动点问题

几何中动点问题的特点是:图形中的动点或动线按某种规律运动,各个动点或动线在运动变化的过程中互相依存,要探求动点或动线运动到何位置时满足一定的"图形条件".……     

Feigenbaum方程的连续可微单峰解*

对著名的Feigenbaum方程.本文建立了它的单峰解的结构定理,由此,得出了寻求单峰解的又一途径.作为例证,本文循此直接去求得了非对称的连续单峰解及C1类的解.     

一类几何流方程周期解的爆破

研究双曲平均曲率流中一类几何流方程周期解的爆破问题.引入合适的黎曼不变量,将该方程化为对角型的一阶拟线性双曲型方程组.该方程组在Lax意义下不是真正非线性的.假设初值是周期的,且在一个周期内全变差很小,此外假设初值还满足一定的结构条件,可以证得该几何流方程的周期解必在有限时间内发生爆破,解的生命跨度估计可以给出.     

The Numerical Solution of Sets of Linear Equations Arising from Ritz-Galerkin Methods

The Ritz-Galerkin solution of a linear integral or differentialequation or set of equations leads to a set of linear algebraicequations, the structure of which depends on the type of expansionset used. For a finite-element expansion, the matrix involvedis sparse, and reasonably efficient solution techniques areknown. We study here the alternative case when a "global" expansionis chosen. Then the matrix involved is in general full, buthas nonetheless a characteristic structure; we discuss the waysin which this structure can be used to yield efficient solutionmethods. Our main result is that a block iterative method canyield an arbitrarily high convergence rate; however, we alsoconsider the stability of a direct solution of the equations.     

Solving Implicit Equations Arising from Adams-Moulton Methods

A new algorithm is given in this paper, which uses functional iteration to solve the implicit equations generated by the Adams-Moulton method. Compared with traditional function iteration, it has three advantages: (1) the center of the circle of convergence for the iteration moves to the left in h plane; (2) the radius of the circle is much enlarged; (3) for a fixed number of iterations, in practice, we can view it as an explicit method since it has a very large absolute stability region. The method is very suitable for stiff system, especially for very large systems.     

A Functional Equation Arising from Simultaneous Utility Representations

Suppose that two classes of utility representations of preferences, one additive and one increasing increments, hold simultaneously over uncertain binary alternatives (gambles). This assumption leads to the functional equation $$ f[h(x-y)+y]=f[h(x)]-f[h(y)]+f(y)\qquad (\kappa >x\geq y\geq 0), $$ and to the inequality h(z) ≤ z (z ∈ [0, κ[), where the functions ? and h are strictly increasing maps of the real interval [0, κ[ onto the real intervals [0, λ[ and [0, μ[, respectively, κ, λ, μ ∈]0, ∞]. We present all solutions under the additional assumption of (first-order) differentiability.     

Integrable Equations Arising from Motions of Plane Curves. II

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9–33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.     

Asymptotic Properties of Solutions of Parabolic Equations Arising from Transient Diffusions

This work is concerned with asymptotic properties of a class of parabolic systems arising from singularly perturbed diffusions. The underlying system has a fast varying component and a slowly changing component. One of the distinct features is that the fast varying diffusion is transient. Under such a setup, this paper presents an asymptotic analysis of the solutions of such parabolic equations. Asymptotic expansions of functional satisfying the parabolic system are obtained. Error bounds are derived.     

A Weighted Simplex Procedure for the Solution of Simultaneous Non-Linear Equations

The Weighted Simplex procedure, a version of the secant algorithm,provides an alternative to the Newton-Raphson procedure in thesolution of a set of non-linear equations. The results of trialsshow that, given a sufficiently good initial approximation toa solution, the two procedures converge with comparable efficiencies.Unlike the Newton-Raphson procedure the Weighted Simplex proceduredoes not involve the computation of partial derivatives.     

由退化抛物型方程产生的常微分方程一种自由边界问题

Assume B∈C[0,+∞)∩C¹(0,+∞) and B(υ) is strictly increasingand concave. That the free boundary Problem for ODE     

板几何中奇异迁移方程解的渐近稳定性

在L~p(1p+∞)空间,研究了板几何中一类具反射边界条件的各向异性、连续能量、均匀介质的奇异迁移方程,通过构造算子,利用比较算子方法,证明了奇异迁移算子A相应的奇异迁移半群V(t)(t≥0)的Dyson-Phillips展开式的一阶余项R_1(t)的紧性,得到了半群V(t)与U(t)(streaming算子B产生)本质谱相同,本质谱型一致;迁移算子A的谱在区域T中由有限个具有限代数重数的离散本征值组成;迁移方程解的渐近稳定性.