级数解

Plentiful and substantial achievements have been made, which motivate a lot of methods to obtain the exact solutions of solitary wave equations, such as: the homogeneous balance method, the hyperbolic function method, the power series method, besides the traditional methods, such as: the inverse scatting method, the bilinear Hirota method, and the Backlund transformation method etc.

求解孤立波方程精确解的方法除了传统的反散射方法、Hirota双线性方法、Backlünd变换方法外，近年来又涌现了很多新方法，如齐次平衡法、双曲正切法，级数展开法等。

Short term solutions could be given by power series, but these were not useful in determining long-term behavior.

短期解可以由幂级数给出，但是这些对于决定长期行为是毫无用处的。

The eigenvalue problem of Dirac operator with perturbation Parameter is studied,the analyticity of perturbation parameter is discussed,and the analytic extension is done,at last,the expansion formulas of power serier about preradical expansion into perturbation parameter are got.

针对带有微扰参数的Dirac算子特征值问题，讨论了对微扰参数的解析性，并进行了解开拓，最后得到了预解核对微扰因子的幂级数展开式。

By use of first-order linear expansion of Taylor series, the linearization process of both nonlinear constraints and objective function of long-term optimal scheduling model are performed and an optimal scheduling model based on seccessive linear programming is built; an approach to solve Optimal scheduling of cascade hydropower station problem by seccessive linear programming is put forward, the scale-down factor of iteration step length is used to ensure that the algorithm can converge to the optimal solution of optimization problem quickly and accurately.

通过采用泰勒级数一阶描述形式，对优化调度目标函数和约束条件中的非线性约束进行线性化处理，建立了基于连续线性规划算法的优化调度数学模型，提出了用连续线性规划技术求解梯级水电站优化调度问题的算法，并采用迭代步长的动态比例缩减因子保证算法能快速准确地收敛到优化问题的最优解。

Avoiding the complex and precise discussion,this article,for a type of elemental heat equation,is going to give a convenient method of solving heat equation by separation of variables and to expand the range of solvable problems with the help of Fourier series.

本文避开微分方程定性理论的复杂、严谨讨论，想就一类最基本的热传导方程，通过分离变量的方法将偏微分方程的边值问题转化为常微分方程的边值问题，给出一种较简捷的定解模式，并利用Fourier级数，将解题范围加以推广。

In this dissertation, many works have been done on the basis of the summarization of the merits of all kinds of meshless methods after a large number of relevant references are read and studied. The basic ideas, principles of meshless methods and their development in the recent years are introduced synoptically. Applications of Element-free Galerkin method in elastic foundation plate are studied deeply and some useful conclusions are obtained as follows:(1) Comparisons between numerical solutions and analytical solutions for several different styles of weight functions, including Gaussian weight function, spline weight function and singular weight function etc, are performed. The results show that Gaussian weight function is the best one.(2) Ranges of relevant parameters for Gaussian weight function are given.(3) Influences of nodal distribution schemes and the number of Gaussian integrated dots on computational results are discussed.

在本文中，作者在认真阅读和研究所搜集到的大量参考文献、综合各种无网格法优点的基础上，首先，对无网格法的基本思想、基本原理、发展现状作了概括性的介绍；其次，对无单元Galerkin法在弹性地基板计算中的应用进行了较为深入的探讨：(1)分别对高斯型、样条型、奇异型等不同形式的权函数进行比较分析，数值计算结果与级数解析解的比较表明，高斯型权函数效果较好：(2)对高斯型权函数进行参数研究，得到地基板计算中相关参数的选取范围；(3)对不同节点分布方案和不同高斯积分对计算结果的影响等作了有益的讨论。

The solution is proved that it can be written as the sum of limited polynomial series.

通过所做的非线性函数变换，得出了考虑雷暴活动的全球电模式的近地稳态解析解，并证明其能展开为有限多项式级数之和。

The solutions of an infinite solid with spherical inclusion under tensile force at infinite are presented.

2利用Legendre级数展开的方法，给出了类位错模型下包含球形夹杂的无限大弹性体受拉伸作用的解。

So, in this paper, a circular cavity witha large radius is used to replace the straight boundary of thehalf space, then the half space problem can be changed tothe scattering problem of two circular cavities to the steadyincident P-wave . Having the aid of the mature cylinderfunction theory, the general solutions of the wave functionscan be given, and an infinite linear algebraic equations ofthe unknown coefficients in the wave functions can be gottenwith the boundary conditions and the Fourier complex seriesexpansion technology, the infinite linear algebraic equationscan be approximately solved by the finite trunction withsatisfying some definite precision, at the basis of thissolution of the equation, the variations and the lay-outs ofthe DSCF at the circular cavity boundary vs. the differentincident angles, the different embedded depths of thecircular cavity as well as the different dimensionless wavenumber of the incident P-wave.

为此，本文采用一个半径很大的圆孔来代替半空间的直边界，将该半空间问题转化为一无限大空间中两个圆孔对稳态P波的共同散射问题，借助于成熟的柱函数理论，通过写出问题波函数的一般形式解，利用问题的边界条件，并采用复数傅立叶级数展开技术将其化为一个仅包含问题波函数中未知系数的一无穷线性代数方程组，在满足一定计算精度的前提下，通过有限项截断进行近似求解，进而讨论了圆孔边界处的动应力集中系数随不同入射角、不同的圆孔掩埋深度、入射波的不同无量纲波数以及介质的泊松比变化和分布情况。

The one is Pagano method which has been proven effective to static problems, but it is very complicate for the three-dimension dynamic problems of finitely long piezoelectric laminated plates.

同时，指出Pagano方法求解动态问题所遇到的困难，为克服这些困难，提出了一种求解压电层合板动态问题三维精确解的幂级数展开方法。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。