Multiple Scale and Singular Perturbation Methods
Multiple Scale and Singular Perturbation Methods
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DOI：
10.1007/978-1-4612-3968-0
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发表时间：
1996-05
期刊：
影响因子：
--
通讯作者：
J. Kevorkian;J. Cole
中科院分区：
文献类型：
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作者：
J. Kevorkian;J. Cole
这本书是一个修订和更新的版本，包括一个新的材料，我们的文字摄动方法在应用数学（施普林格出版社，1981年）的实质性部分。我们目前的材料在一个水平，假设一些熟悉的基础知识，常微分方程和偏微分方程。一些更先进的想法审查需要，因此，这本书可以作为一个文本，无论是在先进的本科课程或研究生水平的课程的主题。微扰法最初被天文学家用来预测小扰动对天体名义运动的影响，现在已经成为几乎所有科学分支中广泛使用的分析工具。如果一个问题”接近”一个可以精确求解的简单问题，那么它就适合于摄动分析。通常，这种接近性是通过在控制系统（由微分方程和边界条件组成）中出现一个小的无量纲参数E来测量的，因此对于E= 0，所得到的系统是精确可解的。所使用的主要数学工具是关于E的函数的适当渐近序列的渐近展开。在正则摄动问题中，一个简单的过程导致一个微分方程系统和渐近展开中每一项的边界条件。该系统可以递归求解，并且对于整个感兴趣域中的所有自变量值，结果的精度随着E变小而提高。第一章讨论正则摄动问题。
This book is a revised and updated version, including a substantial portion of new material, of our text Perturbation Methods in Applied Mathematics (Springer Verlag, 1981). We present the material at a level that assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate-level course on the subject. Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually all branches of science. A problem lends itself to perturbation analysis if it is" close" to a simpler problem that can be solved exactly. Typically, this closeness is measured by the occurrence of a small dimensionless parameter, E, in the governing system (consisting of differential equations and boundary conditions) so that for E= 0 the resulting system is exactly solvable. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of E. In a regular perturbation problem, a straightforward procedure leads to a system of differential equations and boundary conditions for each term in the asymptotic expansion. This system can be solved recursively, and the accuracy of the result improves as E gets smaller, for all values of the independent variables throughout the domain of interest. We discuss regular perturbation problems in the first chapter.