大气运动基本方程组的解析解

在已知大气运动基本方程于光滑函数类中具有最好的稳定性前提下,讨论了它的局部解的解空间构造．根据它的解空间构造,分析了这个方程具有代表性和应用性的第三初值问题,在解析函数类中给出了适定的第三初值问题的解析解的计算方法以及具体的关系表达式,在局部解意义下完整的解决了这一点初值问题的解析解所涉及的理论与计算问题．指出其它类型定解问题都可以仿照文中的计算方法和步骤,求出所需要的稳定的解析解．     

拟流函数法——叶轮机械内三维流动分析问题和设计问题的一种求解方法

本文从动量方程的特点出发并满足连续方程,引入了两个拟流函数.每个拟流函数的主方程均只包含其自身的二阶偏导数,而不包含另一个拟流函数的二阶偏导数.这样,完全的三维解便可通过两个拟流函数主方程的分别单独求解和它们之间的相互迭代来获得.文中给出了在任意非正交坐标系中拟流函数的主方程和相应的边界条件.对叶轮机械内部三维气动分析问题和设计问题的求解进行了讨论和计算,并与解析解、其他数值解法做了比较.结果表明:这种拟流函数方法计算准确而简便,易于得到收敛结果,是求解三维无粘流动的好方法.     

基于MQ拟插值函数逼近的非线性动力系统数值求解

借助多重二次曲面（multi quadrics,MQ）拟插值函数具有较好精确性和稳定性的优势,研究了基于MQ拟插值函数和4阶Runge-Kutta法相结合的方法,构造了求解带有初值问题的非线性动力系统的数值解法,分析了该方法与已有主要方法的优缺点,并给出了相应的数值算例、误差估计.结果表明该方法计算量小、能很好地逼近非线性动力系统的解析解.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。     

求解方程实根个数问题的策略

在国内外数学竞赛中，常有确定方程实根个数的竞赛试题．这类问题比解方程问题更灵活、更深刻、更富思考情趣．本文以最新数学竞赛题为的，充分揭示其常见的求解策略与技巧．1观察分析立竿见影根据方程的特点，从左、右两边的符号、数值特征等入手，进行细致的观察、分析，即可快捷获解．例1（1992年"希望杯"高二第2试试题）方程的实根个数是（）解左边≤2，右边≥2＋1＝3，此方程无实数根．故应选（A）．2求出实根直达彼岸若所给方程易解，则求出其根，立即作答．例2（1995年全国高中数学联赛试题）用[X]表示不大于实数X的最大整数.方程1…     

一类大气尘埃等离子体扩散模型研究

研究了一类大气非线性尘埃等离子体扩散方程初值问题.首先在无扰动情形下,利用Fourier变换方法得到了尘埃等离子体扩散方程初值问题的精确解,接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出了非线性尘埃等离子体扰动初值问题的各次近似解析解.并引用不动点理论,指出了近似解析解的有效性和各次近似解的近似度,通过举例, 用模拟曲线和表格作了近似对照.最后,简述了用同伦映射方法得到的近似解的物理意义.简叙了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

一种求三维弹性力学基本解析解的特征方程解法

提出了一种简单的推导各向同性材料,三维弹性力学问题基本解析解的特征方程解法．应用三维问题控制微分方程的算子矩阵,通过计算其行列式可得到问题特征通解所需满足的特征方程．将满足各种不同简化特征方程的特征通解,代入到微分方程算子矩阵所对应的不同的缩减伴随矩阵,可推导得出相应的三维弹性力学问题的基本解析解,包括B-G解、修正的P-N（P-N-W）解和类胡海昌解．进一步对各类多项式形式的基本解析解的独立性进行了讨论．这些工作为构造数值方法中所需的完备独立的解析试函数奠定了基础．     

基于Shifted Legendre多项式非线性年龄结构种群模型的数值解

基于Shifted Legendre多项式研究非线性年龄结构种群模型的数值解问题.定义了在区间[0,A]×[0,T]上函数的Shifted Legendre逼近多项式,通过Shifted Legendre算子矩阵结合Tau方法,把求解非线性年龄结构种群模型的数值解问题转化成非线性代数方程的求解问题.数值算例的结果显示该算法有效.     

一类广义非线性强阻尼扰动发展方程的行波解

研究了一类非线性强阻尼广义扰动发展方程问题.它们在数学、力学、物理学等领域中广泛出现.首先,引入一个行波变换,把相应的偏微分方程问题转化为行波方程问题并求出原典型问题的精确解.再用小参数方法和引入伸长变量构造了问题的渐近解.最后, 用泛函分析的不动点理论证明了原非线性强阻尼广义扰动发展方程初值问题渐近行波解的存在性,并证明渐近解具有较高的精度和一致有效性.该文求得的渐近解是一个解析展开式, 所以它还可继续进行解析运算, 而单纯用数值模拟的方法是不行的.     

激光脉冲放大器增益通量耦合系统解

研究了一个激光脉冲放大器增益通量系统解的问题.首先讨论了较一般的系统, 然后引入一个同伦映射.再利用映射的性质, 引进一个人工参数, 将求解非线性问题转化为求解一系列线性问题.再逐次地求出对应的线性问题的解, 最后得到了原模型解的近似展开式.可以看出, 同伦映射方法是一个解析的方法.它是通过函数的解析运算并用初等函数来表达近似解,其不同于用离散数值运算的数值计算方法.因此通过同伦映射解, 还可以对它继续进行解析运算, 从而可以进行微分和积分等运算来得到与激光脉冲放大器增益通量相关的其他物理量的性态.     

High-order nonlinear boundary value problems admitting multiple exact solutions with application to the fluid flow over a sheet

We frame a hierarchy of nonlinear boundary value problems which are shown to admit exponentially decaying exact solutions. We are able to convert the question of the existence and uniqueness of a particular solution to this nonlinear boundary value problem into a question of whether a certain polynomial has positive real roots. Furthermore, if such a polynomial has at least two distinct positive roots, then the nonlinear boundary value problem will have multiple solutions. In certain special cases, these boundary value problems arise in the self-similar solutions for the flow of certain fluids over stretching or shrinking sheets; examples given include the flow of first and second grade fluids over such surfaces.     

Symmetric Boundary Value Methods for Second Order Initial and Boundary Value Problems

We introduce symmetric Boundary Value Methods for the solution of second order initial and boundary value problems (in particular Hamiltonian problems). We study the conditioning of the methods and link it to the boundary loci of the roots of the associated characteristic polynomial. Some numerical tests are provided to assess their reliability. Dedicated to the memory of Professor Aldo Cossu     

A “Discretization” Technique for the Solution of ODEs II

A functional analytic technique was recently presented for finding discrete equivalent counterparts of initial value problems of ODEs and obtaining their real analytic solutions. In the current paper, this technique is extended to boundary value problems of ODEs and to the complex solutions of ODEs. In order to demonstrate this technique, it is applied to the classic Blasius problem of fluid mechanics. Apart from its real solution, its complex solution is also studied. The obtained results indicate that the complex Blasius function exhibits an oscillatory behavior and strengthen a conjecture regarding its singularities in the complex plane.     

On One Method for Factorization of Algebraic Polynomials

We propose a method for the factorization of algebraic polynomials with real or complex coefficients and construct a numerical algorithm, which, along with the factorization of a polynomial with multiple roots, solves the problem of the determination of multiplicities and the number of multiple roots of the polynomial.     

求解多项式规划的一个全局最优化算法

提出一个求解带箱子约束的一般多项式规划问题的全局最优化算法, 该算法包含两个阶段, 在第一个阶段, 利用局部最优化算法找到一个局部最优解. 在第二阶段, 利用一个在单位球上致密的向量序列, 将多元多项式转化为一元多项式, 通过求解一元多项式的根, 找到一个比当前局部最优解更好的点作为初始点, 回到第一个 阶段, 从而得到一个更好的局部最优解, 通过两个阶段的循环最终找到问题的全局最优解, 并给出了算法收敛性分析. 最后, 数值结果表明了算法是有效的.     

On the baire classification of Sergeev frequencies of zeros and roots of solutions of linear differential equations

We show that the upper and lower characteristic frequencies of zeros and the upper frequency of roots of a solution of a linear differential equation treated as functions on the direct product of the space of equations with the compact-open topology by the space of initial vectors of solutions belong to the third Baire class and that the lower characteristic frequency of roots belongs to the second Baire class. As a corollary, we show that the ranges of the considered frequencies on the solutions of a given equation are Suslin (analytic) sets. In addition, we prove the Lebesgue measurability and the Baire property of the extreme characteristic frequencies of zeros and roots of an equation treated as functions of a real parameter on which the coefficients of the equation depend continuously.     

A family of multistep methods to integrate orbits on spheres

Summary A trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution, rather than in finding the actual correspondence between the values of the parameter and the points on that curve. This paper introduces a family of multi-stage, multi-step numerical methods to integrate trajectory problems whose solution is on a spherical surface. It has been shown that this kind of algorithms has good numerical properties: consistency, stability, convergence and others that are not standard. The latest ones make them a better choice for certain problems.     

A Class of Nonlinear, Nonhyperbolic Systems of Conservation Laws with Well-Posed Initial Value Problems

With mild restrictions on the initial data, we show well-posedness of the initial value problem for systems of conservation laws in one space variable with real and equal characteristic speeds and a deficiency of one corresponding eigenvector. The obtained weak solutions assume values as Borel measures at each time after a smooth solution ceases to exist. The concept of entropy density-flux functions is extended to such systems. Finally, we show that systems with well-posed initial value problems, admitting weak solutions of this type, necessarily satisfy these structural assumptions.     

Propagation dynamics of a diffusing substance on the surface and in the bulk of water

Two- and three-dimensional problems of propagation of a diffusing substance on the surface and in the bulk of water are considered. An analytic solution to boundary value problems for the diffusion equation is proposed in unbounded domains for the initial condition of special form. The above-threshold range of concentrations of the diffusing substance is analyzed. Propagation of the diffusing substance along the free surface and at the bottom of a basin is considered. Analytic solutions to the problems are obtained by the Fourier method followed by the expansion of an arbitrary function in terms of Bessel functions and Legendre polynomials. The analytic solutions constructed are compared with the numerical solutions of a boundary value problem obtained by the software package Mathematica. The size of the pollution spot as a function of time, as well as the effect of geometric and physical parameters used on the spot size, is analyzed. The mathematical models considered have an important applied value in the problem of environment protection in case of emergency situations on ships.     

On a free interface problem for linear ordinary differential equations and the one-phase Stefan problem

A numerical method for the solution of the one-phase Stefan problem is discussed. By discretizing the time variable the Stefan problem is reduced to a sequence of free boundary value problems for ordinary differential equations which are solved by conversion to initial value problems. The numerical solution is shown to converge to the solution of the Stefan problem with decreasing time increments. Sample calculations indicate that the method is stable provided the proper algorithm is chosen for integrating the initial value problems.