Taylor级数的解析延拓与一类非线性常微分方程的两点边值问题
Analytic continuation of Taylor series and the two-point boundary value problems of some nonlinear ordinary differential equations
摘要：
本文比较和讨论了分别基于Taylor（Maclaurin）级数、Padé逼近和保角映射的三种方法（每种方法有两种变体）对非线性常微分方程（ODE）两点边值问题的拟解析解的效率.根据常微分方程和边界条件的难度，依次考虑了六种不同的常微分方程和边界条件。在指出泰勒级数方法几乎总是需要求助于解析延拓程序才能有效之后，我们使用剩下的两种方法（Padé和保角映射）的互补性来说明它们各自的优点和局限性。我们强调的重要性，存在的解决方案与移动奇点的方法的效率，特别是所谓的Padé-Hankel方法。(We表明后一种方法相当于把一个可动极点推到无穷远。）对于每种配置，我们确定的奇异性分布（在复杂的平面上的独立变量）的解决方案寻求，并显示这种分布如何控制这两种方法的效率。一般来说，基于Padé近似的方法易于使用且鲁棒，但在某些情况下可能会很笨拙，而保角映射方法是一种非常精细的方法，当需要高精度时应使用。
英文摘要：
We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Padé approximants and conformal mappings, for solving quasi-analytically a two-point boundary value problem of a nonlinear ordinary differential equation (ODE). Six configurations of ODE and boundary conditions are successively considered according to the increasing difficulties that they present. After having indicated that the Taylor series method almost always requires the recourse to analytical continuation procedures to be efficient, we use the complementarity of the two remaining methods (Padé and conformal mapping) to illustrate their respective advantages and limitations. We emphasize the importance of the existence of solutions with movable singularities for the efficiency of the methods, particularly for the so-called Padé-Hankel method. (We show that this latter method is equivalent to pushing a movable pole to infinity.) For each configuration, we determine the singularity distribution (in the complex plane of the independent variable) of the solution sought and show how this distribution controls the efficiency of the two methods. In general the method based on Padé approximants is easy to use and robust but may be awkward in some circumstances whereas the conformal mapping method is a very fine method which should be used when high accuracy is required.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
