arXiv 1111.71492011-11math-phmath.MP

微分变换方法的现状

Status of the differential transformation method

C. Bervillier

备注：: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations added
发表信息：: Appl. Math. Comput. 218: 10158,2012
DOI：: 10.1016/j.amc.2012.03.094

摘要：

进一步到最近的争论是否微分变换方法（DTM）解决微分方程是纯粹的和唯一的传统的泰勒级数方法，它强调的是，DTM目前使用，往往只是作为一种技术（解析）计算幂级数的解决方案（在初始值参数方面）。有时，分段解析延拓过程在数值例程（例如，在射击方法内）或在半分析程序中（例如，解决边界值问题）。还强调的是，在其发明的时候，DTM的当前使用的基本成分（将微分方程转换为可迭代求解的相同阶的差分方程）已经被“传统”泰勒方法的用户（特别是在软件包的详细说明中-数值例程-用于自动求解常微分方程）知道了很长一段时间。现在，DTM的捍卫者仍然忽视了对“传统”泰勒方法用户的研究，尽管发展得更好，反过来，他们似乎也同样忽视了DTM的存在。DTM被赋予了明显的强形式化（与傅立叶、拉普拉斯或梅林变换相同的基础）。虽然经常使用琐碎，它是容易实现的，并容易适应不同种类的微分程序。这使得它非常有吸引力。因此，泰勒方法在各种问题上的应用，以及更一般的幂级数方法（包括非整数幂）已经被勾画出来。它的潜力似乎没有得到应有的开发。在讨论了引起争论的“误解”的原因之后，对前面的问题进行了具体的说明。

英文摘要：

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.

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