目录
5 60 本教材点评
5.4 常微分方程 6 本
《A First Course in the Numerical Analysis of Differential Equations》
作者：Arieh Iserles
出版商：Cambridge University Press
出版年：2008
ISBN：9780521556552
适用范围：高年级本科生，研究生
推荐强度：8
作者简介：Arieh Iserles（生于1947年9月2日）是一位计算数学家，现为剑桥大学微分方程数值分析教授，并且是应用数学和理论物理系成员。主要研究方向：常微分方程和偏微分方程，逼近理论，几何数值积分，正交多项式，功能方程，计算动力学和高度振荡现象的计算。他是Acta Numerica的执行编辑，《 IMA数值分析杂志》的主编以及其他数本数学杂志的编辑。从1997年到2000年，他担任计算数学基础学会主席。2010-2015年，他担任剑桥分析中心（CCA）的董事；1999年，他被挪威科技大学授予Onsager奖章；2012年，他获得了大卫·克莱顿勋章，由数学及其应用研究所和伦敦数学协会颁发，以表彰他在数学和数学上的服务， 并在2014年被工业和应用数学协会授予SIAM专业杰出服务奖；2012年，Iserles教授应邀在2012年7月2日至7日在克拉科夫举行的第六届欧洲数学大会上 发表演讲。
书评：
Aimed at advanced mathematics students, this book is a text whose goal is to introduce students to numerical analysis and computation of differential equations. The book avoids the traditional “definition—theorem—proof”style of most mathematics texts at its level, but it still requires significant mathematical sophistication on the part of the reader. Mathematical questions that arise in numerical analysis of differential equations are emphasized while not ignoring the practical aspects putting the theory to work. The book is successful in carving out a niche that is distinctly different from other books with which it might be compared. Thus this is not just another mathematical introduction to differential equations, a “cookbook” of standard methods for solving differential equations, or an advanced treatise that gives a comprehensive set of alternative methods for solving a particular class of problems. Instead, the book starts from the vantage point of typical advanced undergraduate or beginning graduate student interested in exploring scientific computation and presents material that introduces many of the issues that arise in the creation of effective algorithms to solve differential equations.
The book has three sections devoted to ordinary differential equations, th Position equation, and evolution equations. Each section has a somewhat different feel and stresses different aspects of the interaction between mathematics and computation. The first section introduces multistep and Runge-Kutta algorithms for solving initial value problems. It then devotes chapters to still systems and to error control for adaptive step-size algorithms. The section ends with a chapter that discusses iterative solutions of systems of equations by Newton’s method and it’s variants. In comparing this section Iserles’ text with the classic text of Henrici, one finds here lots of material based on research during the intervening years. This reflects the strength of this book in bringing the experience gained from the past 50 years of scientific computing into a setting aimed at mathematics students. I applaud Iserles for Lowering the barriers between “pure” and “applied” mathematics by blending this computing experience into the mathematics.
The second section of the book, on solutions of the Poison equation, emphasizes numerical linear algebra. Following chapters that introduce boundary value problems for ordinary differential equations and the finite-element method for solving the Poison equation, the section devotes three chapters to methods for solving sparse systems of linear equations and one to a discussion of fast Poisson solvers.
The theme of the third section is stability of algorithms for solving initial value problems for evolution equations. There are only two chapters, and the author conveys his feeling that the material he discusses is hardly a “finished” subject. There are excellent illustrations of the consequences of instability and the essential differences between solving initial value problems for ordinary and partial differential equations.
As a mathematician who developed an interest in numerical analysis in the middle of his professional career. I thoroughly enjoyed reading this test. I wish this book had been available when I first began to take a serious interest in computation. The author’s Style is comfortable, and I enjoyed the way in which he expresses his personality in much of the material. As much as I like the book, I suspect that it may have difficulty attracting a sizable following. Much of the growth in scientific computation is occurring in disciplines where there is less concern for the logical structure and mathematical analysis of numerical methods than is reflected in this book. These audiences are likely to prefer texts that emphasize practice and are less demanding of mathematical background of students and teachers. That would be unfortunate because the insight that comes from understanding the mathematics underlying algorithms for solving differential equations is an excellent foundation for scientific computation. One could easily argue that the material in this book is far from important today than much of that taught in courses that introduce mathematics majors to applied mathematics. This book would be my choice for a text to “modernize” such courses and bring them closer to current practice of applied mathematics.
本书旨在向高年级本科生与研究生介绍数值分析与微分方程的计算。它没有采用同层次大多数数学教材“定义-定理-证明”的传统方式，却也需要读者拥有扎实的数学基础。本书格外强调微分方程数值分析中的数学问题，同时也没有忽视实际应用。本书成功开辟了一条与其他教材完全不同的道路。也正因此，本书不仅仅是一本介绍微分方程的数学教材，一本微分方程标准解法的“食谱”，或者一篇为一类特定问题提供几乎所有解法的专题论文。相反地，本书从高年级本科生或低年级研究生探索科学计算的视点出发，介绍了构思微分方程高效算法时出现的种种问题。
全书分为三部分，分别介绍了常微分方程、Poisson方程与演化偏微分方程。每部分都给人不同的感觉，强调了数学与计算间相互作用的不同方面。第一部分介绍了求解初值问题的多步法和Runge-Kutta算法。本书接下来花费了一些章节来讲述刚性系统与自适应步长算法的误差控制。这部分以对方程组的牛顿法和其变体的讨论作结。与经典的Henrici的教材中相应部分相比，你可以在本书的此部分中发现许多基于近些年研究的内容（译者注：本书出版于2008年，而Peter Henrici的著作大多出版于上世纪80年代）。这反映了本书的力量，它把过去50年科学计算的经验引入到一个面向数学专业学生的环境中。我十分赞许Iserles的工作：通过将实际科学计算的经验引入数学系统，来拉近基础数学与应用数学间的距离。
本书的第二部分，有关Poisson方程的求解，强调了数值线性代数的重要性。这部分介绍了常微分方程的边值问题与Poisson方程的有限元方法，花了三章来介绍稀疏线性方程组的解法，并花了一章来讨论快速Poisson解算器。
本书第三部分的主题是演化方程初值问题求解算法的稳定性。这部分只有两章，并且作者想表达的是，这些内容很难称得上是一个知识体系完备的学科。这两章提供了一些不错的例子，来说明方程不稳定的后果，以及偏微分方程和常微分方程初值问题的本质区别。
作为一名在职业生涯中段开始对数值分析产生兴趣的数学家，我在阅读本书时十分享受。我也希望在我真正开始研究计算数学时本书能派上用场。这本书读起来不怎么费力，同时我也很享受他在书中展现独特观点的方式。尽管我很喜欢这本书，但我怀疑它可能很难吸引大批读者。因为科学计算飞速发展的学科并不像本书所反映的那样，将很多关注点放在了数值方法的逻辑结构和数学分析上。那些读者，包括学生和教师，可能更喜欢强调实际应用而不需要太多数学背景的教材。然而很不幸运，通过理解求解微分方程的数学算法，提升洞察力，才能形成扎实的科学计算基础。人们可能会认为，相比如今那些将数学专业的学生引入应用数学领域的课程而言，本书的内容早已不那么重要。但对我来说，如果以“现代化”这门课程，并以贴近应用数学前沿实践为目的，这本书将是绝佳之选。
点评人： John Guckenheimer（康奈尔大学应用数学中心主任，数学系教授）
资料整理：方诗雨
翻译：赵旭彤
《Ordinary Differential Equations》
作者：Vladimir I. Arnol’d
出版商：Springer
出版年：1992
ISBN：9783540548133
适用范围：高年级本科生，研究生
推荐强度：9
作者简介：Vladimir I. Arnol’d（俄语：Влади́мир И́горевич Арно́льд，1937年6月12日－2010年6月3日），俄国数学家，生于苏联敖德萨（今乌克兰境内）。1957年他19岁时就解决了希尔伯特第十三问题，此后对多个数学领域都有重大贡献，包括动力系统理论、突变论、拓扑学、代数几何、经典力学、奇点理论。他最著名的成果是关于可积哈密顿系统稳定性的KAM定理，即柯尔莫哥洛夫-阿诺德-莫泽定理。他的学术成就深得肯定，获颁多个奖项，如1982年的克拉福德奖，2001年的沃尔夫数学奖，2008年的邵逸夫奖等。
书评：
Vladimir Arnol’d is a master, not just of the technical realm of differential equations but of pedagogy and exposition as well. His Ordinary Differential Equations, now in its third edition, is a classic. Arnol’d writes that he has attempted to limit the ideas he presents to a bare minimum. The heart of his treatment consists of two central notions. The first is the theorem on the rectification of a vector field (from which he extracts existence, uniqueness and differentiability results); the second is the theory of one-parameter groups of linear transformations, from which springs the theory of autonomous linear systems.
Arnol’d focuses a good deal more on the applications of ordinary differential equations to mechanics than comparable texts. This is due in part to the breadth of his interest and expertise in this area but also to the sense that applications to mechanics put meat on the bare bones of the subject. The equation of the pendulum appears early and the author returns to it regularly to illustrate other concepts. So, for example, the conservation of energy appears as an example of first integrals, the small parameter method is derived as a consequence of the theorem on differentiation with respect to a parameter, and the study of the swing of the pendulum is naturally connected to the theory of differential equations with periodic coefficients.
Arnol’d also emphasizes the geometrical and qualitative aspects of the subject more than other authors. In support of this, he includes an abundance of small figures embedded in the text. He introduces phase space and phase flows early on and makes extensive use of these concepts.
The book is divided into five chapters: basic concepts (phase space, vector fields, phase flows), basic theorems (the rectification theorems and their consequences), linear systems (including a careful treatment of the matrix exponential), proofs of the main theorems, and differential equations on manifolds. The writing throughout is crisp and clear.
In this third edition, the first and second chapters have been substantially revised. There are new sections on elementary methods of integration for homogeneous and inhomogeneous first order linear equations, as well as on first order linear and quasi-linear partial differential equations. The author has also added material on Sturm’s theorems on the zeros of second order linear equations. In this new material the author once again focuses on the geometric core of the methods he describes and ties the results to applications, particularly in mechanics.
Arnol’d says that the book is based on a year-long sequence of lectures for second-year mathematics majors in Moscow. In the U.S., this material is probably most appropriate for advanced undergraduates or first-year graduate students.
An English translation (by a different translator) of the first edition of this book was originally published by the MIT Press in 1973; a revised edition appeared in 1978, And seems to be still in print. It has the same title as the book under review.
Vladimir Arnold 是一位大师，不仅仅是在微分方程的技术领域，同样也在微分方程的教学与阐述上。他的《Ordinary Differential Equations》，如今已是第三版，是当之无愧的经典之作。Arnold写到，他试图将他的想法简化到最低限度。他论述的核心包括两个中心概念。第一个是向量场的直化定理（他从中得到了关于存在性、唯一性与可微性的结果）；第二个是单参数线性变换群的理论，他也由此发展出了自治线性系统理论。
相比其他教材，Arnold更关注常微分方程在力学上的应用。这不仅仅因为他对这个领域有着浓厚的兴趣与大量的专业知识，同样也因为对常微分方程来说，力学上的应用将这门学科从一具骨架变成了真正的人。作者先引入钟摆方程，再将它一般化来展示其他概念。所以，举个例子，能量守恒作为首次积分的例子出现，小参数法也由对参数的可微性定理导出，并且关于钟摆的研究也会很自然地与周期系数微分方程的理论联系在一起。
相比其他作者，Arnold也更加强调常微分方程几何与定性的方面。为了论证这些，他在书中插入了大量图片。他在本书的开始就介绍了相空间和相流，并且也确实在很多地方用到了这些概念。
本书被分为五章：基础概念（相空间、向量场、相流），基本定理（直化定理及其推论），线性系统（包括有关矩阵指数运算的详细叙述），基本定理的证明，流形上的微分方程。写作风格简洁明了。
本书的第三版大幅度修改了前两章，包括一些新的小节来叙述齐次与非齐次一阶线性方程中积分的基本方法。作者也同样在叙述二阶线性方程前新增了Sturm定理的内容。在新增的内容中，作者再次专注于解法的几何本质与实际应用，尤其是力学方面。
Arnold曾说本书的编写是基于莫斯科大学数学系二年级本科生的为期一年的课程。而在美国，这些内容更适合高年级本科生或者一年级研究生。
本书第一版的英译版（由另一位译者翻译）由麻省理工学院出版社于1973年出版；修订版于1978年出版，似乎至今仍然在印刷。它的书名与本书相同。
点评人：William J. Satzer（美国明尼苏达大学数学系博士）
资料整理：方诗雨
翻译：赵旭彤
《Elementary Differential Equations and Boundary Value Problems》
作者：William E. Boyce, Richard C. DiPrima
出版商：Wiley
出版年：2008
ISBN：9780470383346
适用范围：高年级本科生
推荐强度：9
作者简介：William E. Boyce（1930-2019），伦斯勒理工学院名誉教授，博伊斯于1987年加入TAA，并于去年成为终身荣誉会员。他拥有罗得斯大学数学学士学位，硕士学位和博士学位，卡内基-梅隆大学数学专业学位，曾任数学系研究生委员会主席和应用数学理学硕士课程协调员。1993年，他成为伦斯勒（Rensselaer）的爱德华·汉密尔顿（Edward P. Hamilton）科学教育杰出教授，退休后被授予爱德华·汉密尔顿（Edward P. Hamilton）荣誉退休教授的头衔。
Richard C. DiPrima(1927-1984)，伦斯勒理工学院数学科学系教授，并于1972年开始担任系主任。毕业于卡内基梅隆大学，曾获古根汉奖学金自然科学类（美国及加拿大），曾在SIAM担任多年职务，并于1979-1980年担任SIAM主席。
书评：
This edition of this book focuses on the theory and practical applications of Differential Equations to engineering and the sciences. Emphasis is placed on the methods of solution, analysis, and approximation. Use of technology, illustrations, and problem sets help readers develop an intuitive understanding of the material. Historical footnotes trace the development of the discipline and identify outstanding individual contributions. This book builds the foundation for anyone who needs to learn differential equations and then progress to more advanced studies. Especially for undergraduate students of mathematics, science or engineering, who typically take a course on differential equations during their first or second year of study. In addition, many problems in this book are entirely straightforward, but many others are more challenging, and some are fairly open-ended, and can serve as the basis for independent student projects. I believe that the most outstanding Feature of this book is the number, and above all the variety and range.
For me, I had a shaky foundation in calculus, particularly infinite series, heading into my differential equations class. It didn't particularly matter with this book. The book reviews a few of the more challenging concepts from calculus when needed and is gentle when explaining new material. Generally speaking, if you can differentiate and integrate and quickly google whatever gaps in knowledge you might have, this is a great book. I can honestly say I've never had a clearer math text. This book really benefits me a lot and solve lots of problems I do not understand before. It occasionally borders on being too thorough by being weirdly repetitive, but this is confusing only until you catch up to the style of the book. I've never had a problem with the examples worked; unlike other math texts, this book uses multiple examples applicable to a variety of situations and with varying difficulty, meaning less of a gap between the difficulty of the examples and the difficulty of the exercise problems. Some chapters are a little less clear and easy to follow than others, thanks to a strict adherence to formality in the derivations. Most of the chapters that I used were fantastic, but as I didn't use the entire book, I can't make any guarantees for all of the topics it covers. The book doesn't take you through derivations that would be too difficult or lengthy for the scope of the book and is clear about why it skips a derivation when it has to, which is much appreciated for saving time and stress.
The grades I made in my differential equations class had a lot of room for improvement, but just taking notes on the text of each chapter gave me roughly a 50% understanding of the material as applied to any problem, which is pretty stellar; once I started running through the exercise problems, I had few difficulties in the class.
It's probably a little weird to say, but I actively enjoyed using this textbook. I had some math anxiety from previous calculus classes but I had no issues learning from this book. It's definitely staying on my shelf.
本书的此版重点关注工程与科学中微分方程的理论和实际应用，尤其强调求解、分析与估计的方法。 本书运用不限于插图与习题的各种方法，来帮助学生建立起对微分方程的直观理解。注脚大多是历史性的，即追溯了学科的发展，并展现了一些杰出学者的贡献。对想要学习微分方程的学生，特别是那些通常在本科前两年修习微分方程课程的理工科学生，本书为其建立起扎实的基础，同时也放眼于更深层次的研究。此外，本书的习题大多简单直接，但也有很多富有挑战性的题目，甚至有一些仍是开放性问题，可以作为学生研究课题的基础。本书的范围很广，涉及的理论与问题种类繁多，这也成为了本书最突出的特点。
当我上微分方程课时，我的微积分基础并不牢固，特别是在无穷级数方面。但对本书来说则无关紧要。本书会在需要时复习与微积分相关的概念，并对新出现的不易理解的内容做出通俗易懂的解释。总而言之，如果你掌握了基础的微分与积分，并能快速地谷歌一下来找到你微积分知识上的断层与缺失，那本书就是相当不错的读物了。说实话，我再也没有读过一本比本书解释得更清楚的教材了。本书真的令我受益良多，并且解决了许多之前我不理解的问题。本书偶尔会因为一些奇怪的重复而变得冗长，但在你理解了它的叙述风格后便不再感到困惑。我从未在本书的例题中遇到过麻烦；不同于其他数学教材，本书提供了大量的例题，涉及各种情形与难度，而这意味着习题与例题间更小的难度差距。有些章节会因追求过于严密的证明而不易理解。我读到的大部分章节很棒，但也因我没读完整本书，我不能保证本书的每章都像我说的那么好。本书没有对于其知识范围来说太难或太长的推导，并且清楚地说明了为何有时要跳过推导，这对节省时间和精力是很有帮助的。
我微分方程课程的成绩仍有很大的提升空间，但仅仅通过在书的每一章中做笔记，我就理解了所学内容的一半以上，而且也确实能用这些知识解决问题，这是相当出色的；当我开始做习题时，我在课上已经遇不到什么困难了。
可能这样说有些奇怪，但阅读本书确实是一种享受。在我之前上微积分课程时，我曾对学习感到些许焦虑，但阅读本书时则没有任何问题。如此优秀的教材，怎么可能不在我的书架上呢？
资料整理：方诗雨
翻译：赵旭彤
《Mathematical Methods for Physics and Engineering》
作者：K. F. Riley, M. P. Hobson and S. J. Bence
出版商：Cambridge University Press
出版年：2006
ISBN：9780521679718
适用范围：高级本科生
推荐强度：10
作者简介：K. F. Riley在剑桥大学获得数学博士学位。是布鲁克黑文大学基本粒子物理学研究助理，后在剑桥卡文迪许实验室（Cavendish Laboratory）担任讲师。他参与了许多早期重子共振的实验发现，曾在克莱尔学院担任高级讲师，在这所学院教授物理和数学已逾40年。他还是《 200个令人困惑的物理学问题》的作者之一。
M. P. Hobson在剑桥大学学习自然科学，专攻理论物理，并留在卡文迪许实验室（Cavendish Laboratory）完成博士学位。他是剑桥三位一体研究室的研究员，后来是粒子物理与天文学研究委员会的高级研究员，他目前是卡文迪许实验室的大学读者，他的研究兴趣包括宇宙学的理论和观察方面，是《广义相对论：引言》的主要作者。
S. J. Bence获自然科学学士学位和博士学位，剑桥大学天体物理学博士学位。之后成为恒星形成过程和恒星形成区域的结构方面研究助理。
书评：
Mathematical Methods for Physics and Engineering by Riley et al is a great scientific textbook. The reader should not be intimidated by its size - over 1200 pages - and skimming through it, one is confronted with a vast amount of information (which could be daunting for students). However, when reading it in detail one realizes that the number of pages is justified and the result is a self-contained reference on mathematical methods for physics.
The book is well organized: it starts at basic university undergraduate level and progresses to the highest graduate level. It is a tour de force to write mathematical sections that are both complete and at an appropriate academic level; the sequence of sections (and their content) has been chosen with great care in order that each section may introduce the following ones. The authors have clearly succeeded with this challenge, making this a remarkable pedagogical book. No precise knowledge is needed at the beginning, and all the material is presented in a logical progression. The student can confidently follow the material as it is presented. Even though this book is devoted to physics and engineering, it can be used by students in other scientific fields.
The material covered in the book concentrates on applied theorems and formulae. But as the title of the book indicates, it is primarily written 'for physics and engineering' so this is to be expected and it greatly simplifies the text of the course. Consequently, derivations of the main results are essentially 'for physicists'. Frequently proofs are briefly outlined and are not as detailed as many mathematicians would like. Nevertheless, the final results are always clearly stated together with the technical conditions under which they should be used, thus providing a very comprehensive and practical textbook. The only criticism is that the sections on probability and statistics do not fit comfortably between their previous and following sections.
The choice of exercises is excellent and possibly the best feature of the book. Firstly, many of them are interesting and original and, secondly, the answers and hints are generally well detailed and explained. Numerous exercises, with full solutions, are included in the text to serve as examples of new concepts or results. In addition, several exercises on longer problems are divided into several small components which are solved independently. In this way, interesting and complex questions are tackled in addition to the usual standard short exercises that are normally found in textbooks on the subject.
This is an excellent book for a three-year university course, but even such a comprehensive book has its limitations. In the introduction the claim that 'the text should provide a useful reference for research workers' is an exaggeration as the text does not extend to this level. In particular, there is a lack of reference to more specialized books in the field. This is not a problem at student level since the book is self-contained and exhaustive, but the scientific researcher could, on occasion, find the information contained in the book inadequate and find no suggestions for further reading.
In summary, this textbook is a great reference at undergraduate levels, particularly for those who like to teach (or to learn) using lots of examples and exercises.
Riley等人所著的《Mathematical Methods for Physics and Engineering》是一本不错的科学教材。读者不应被它那超过1200页的厚度所吓倒，简单地浏览后，你面对的是大量的信息（这可能会让学生望而生畏）。然而，当仔细阅读它时，你会意识到如此多的页数是合理的。也正因那令人生畏的页数，本书成为有关物理学中数学方法的最详尽的参考书之一。
本书编排很好：从适于低年级本科生的内容开始，逐渐提升至高年级研究生的层次。写出完整又有恰当学术水平的数学教材就像一次旅行；为了使每部分都能过渡到后面的内容，章节的顺序（及它们的内容）需要经过深思熟虑。本书的作者们显然在这方面很成功，使它成为一本了不起的教材。开始的部分不需要特定知识，所有内容都按逻辑顺序呈现。即使这本书是专注于物理和工程的，它也可以被其他领域的学生使用。
本书重点关注定理与公式的应用。但正如书名所示，它主要是为“物理和工程”而写的，所以（重点关注定理与公式的应用）是意料之中的，这大大简化了课程的内容。因此，主要结论的推导更贴近物理学家的思考方式。通常证明只被简单地概述，而不像许多数学家所期望的那样详尽。尽管如此，结论的适用条件还是被详细地给出，从而形成了一本既全面又实用的教材。唯一的缺点是有关概率统计的章节与前后内容不太协调。
本书最突出的特点就是其出色的习题。首先，大部分题目兼顾趣味性与独创性，其次，答案和提示通常很详细。教材包含了大量的习题和解答作为新概念或结论的范例。此外，一些较长的习题被拆分成几个可以独立解决的小部分。通过这种方式，除了那些能在其他教材中找到的标准练习外，学生们还可以尝试解决一些有趣而又复杂的问题。
对于三年制大学课程来说，本书是极佳的选择，但即使这样全面的书也有其局限性。在引言中，“本书应当为研究者提供有价值的参考”的说法过于夸张了，因为本书没有达到这个层次，特别是缺乏对同领域更专业书籍的参考。对学生来说这可能不是问题，因为这本书已经足够详尽，但科研工作者有时会发现某些内容仍不够深入，却找不到进一步的阅读建议。
总之，对本科生，特别是那些喜欢通过大量例子与习题来学习的学生而言，本书是一本极佳的教材。
点评人：R. Botet（巴黎-萨克雷大学物理系教授）
资料整理：方诗雨
翻译：赵旭彤
《An Introduction to Ordinary Differential Equations》
作者：James C. Robinson
出版商：Cambridge University Press
出版年：2004
ISBN：9780521533911
适用范围：高年级本科生，研究生
推荐强度：9
作者简介：James C. Robinson，华威大学数学教授，主要研究方向：无限维动力系统；非自治动力系统；将有限维集嵌入欧几里得空间；尺寸理论等。曾获英国皇家工程物理学会领导奖学金（EPSRC Leadership Fellowship）、皇家学会大学研究金（Royal Society University Research Fellowship）、皇家学会联合项目（Royal Society Joint Project）。
书评：
This book is ideally suited to any standard undergraduate course in ordinary differential equations (ODEs) at all levels for mathematics and engineering students. Included in the coverage are both traditional techniques for obtaining explicit solutions to various types of ODEs and an introduction to qualitative methods, especially those associated with phase plane analysis.
Part I and II would be of interest to first year undergraduates. Part I, which concentrates on first order linear and non-linear ODEs, starts at the very beginning with separation of variable techniques. It then proceeds by elaborating upon the standard methods associated with integrating factors, Bernoulli and dimensionally homogeneous types, illuminating the theory with many instructive examples from physics, engineering and population dynamics; the example on the carbon dating of the Shroud of Turin being particularly topical.
Part II explains the standard techniques for solving second order ODEs with constant coefficients, preoccupied as they are with complementary functions and particular integrals.
Part III would be of interest to second year undergraduates with its developments into the realm of second order ODEs with variable coefficients via variation of parameters. These ultimately lead, via Cauchy's equation, into the indicial characteristics associated with the series solution approach, which ultimately generate so many of the interesting special functions like those of Bessel, Airy and Legendre. These topics could run in parallel with those included in Part IV on numerical techniques, which lead from Euler's method into a novel introduction to difference equations. The concepts and techniques of this section are imaginatively illustrated with cobweb diagrams and the like, which help in picturing the ideas associated with fixed points and their stability, periodic orbits and ultimately chaos.
In the midst of all this detail is a very interesting phase plane and portrait coverage of the non-linear oscillations of the simple pendulum, but divorced from the idea of elliptic functions which are not discussed.
Final year students would be interested in Parts V and VI that deal, respectively, with linear and non-linear coupled systems of ODEs.
In Part V matrix techniques for uncoupling linear systems via the modal matrix are introduced with prolific illustration in the phase plane of the role played by the eigen-parameters.
Part VI might only be lightly covered at the undergraduate level, being perhaps more suitable for post-graduates at a deeper level. Here we have a much stronger foray into the quantitative and qualitative analysis of multi-dimensional non-linear systems, leading to the Dulac criterion, the Poincaré-Bendixon theorem and ultimately the Lorentz equations, with their exotic phase plane pictures which have become so topical.
The presentation is sensitive to the needs of students, with careful algebraic steps included in most cases. Appendices on basic required mathematical techniques are also included. The graphical illustrations are mainly drawn with the help of MATLAB.
For a first edition there seem to be surprisingly few misprints. There are trivially some dt's missing in a pair of integrals on page 166.
Some of the figures consisting mainly of single horizontal lines, like Fig. 20.1 on page 177, can easily be overlooked in that, at first glance, they seem to merge with the text. Finally, and perhaps this is more of a problem with MATLAB, some of the line graphs comprising dotted lines, as presented for instance in Figs. 20.5 and 32.7, can be faint enough as to render them almost invisible.
对数学专业与工程专业学生来说，无论他们处于什么水平，只要他们在本科阶段修习常微分方程课程，本书就是一本非常理想的教材。本书既包括求常微分方程解析解的传统方法，也介绍了定性方法，特别是有关相平面分析的方法。
一年级本科生会对第一部分和第二部分感兴趣。第一部分，专注于一阶线性和非线性常微分方程，从最基本的分离变量法慢慢讲起。紧随其后的是对积分因子法、Bernoulli方程与齐次方程的详细说明，伴有大量源自物理、工程和种群动力学的有启发意义的例子；关于都灵裹尸布中碳的年代测定的例子尤为突出。
第二部分解释了求解二阶常系数常微分方程的标准方法，特别是那些与余函数和特殊积分相关的。
第三部分可能会引起二年级本科生的注意，这部分将通过参数变化过渡至二阶变系数常微分方程的领域。通过Cauchy方程，本部分最终会导向与幂级数方法相关的指标特征，并导出一些有趣的函数，如Bessel函数、Airy函数和Legendre多项式。这些内容平行于第四部分中从Euler方法到差分方程的数值技巧。这部分用蛛网图（一种动力系统领域研究一维迭代函数定性性质的可视化方法）来展示相关概念和方法，这对解释与不动点、稳定性、周期轨道和混沌有关的概念有很大的帮助。
在这之中，有一个非常有趣的例子。它与单摆非线性摆动的相平面和相图相关，却避开了未经讨论的椭圆函数的概念。
高年级本科生会对第五与第六部分感兴趣，它们分别与常微分方程中线性和非线性的耦合系统相关。
第五部分介绍了通过模态矩阵求解线性耦合系统的方法，配以许多相平面图来说明特征参数的作用。
第六部分对本科生来说可能只适合稍加介绍，它也许更适合专精于此领域的研究生。在这部分，本书对多维非线性系统的定量与定性分析进行深入的研究，导出了Dulac准则，Poincaré-Bendixon定理与Lorentz方程，并配以一些相平面图。这些内容如今已成为相当热门的研究课题。
本书的内容设计十分体贴，大多数情况下包含了极详细的数学推导。附录涵盖了所需的基本数学知识。书中的图片大多数是在MATLAB的帮助下画出。
第一版的印刷错误似乎少之又少，仅在166页的积分表示中少写了一些“dt”.
有些图甚至只有一条水平线，如177页的图20.1，乍一看，它们似乎与文本交织在一起，因此很容易被忽略。最后，一些包含虚线的图片会因为有些模糊而难以看清，如图20.5和图32.7，不过这更像是MATLAB的问题。
点评人：J. M. H. Peters（利物浦理工学院工程系和数学系）
资料整理：方诗雨
翻译：赵旭彤
《Differential and Integral Equations》
作者：Peter. J. Collins
出版商：Oxford University Press
出版年：2006
ISBN：9780198533825
适用范围：高年级本科生
推荐强度：8
作者简介：Peter J. Collins，St Edmund Hall高级研究员，牛津数学研究所拓扑分析研究组负责人，曾任该大学数学科学学院董事会主席兼该校总董事会副主席。于1966年至1971年担任数学研究讲师，并于1968年至1971年担任圣埃德蒙堂的初级研究员。从1971年至2001年，他担任数学研究员和导师，然后在2002年成为高级研究员（直到2012年）。
书评：
In 362 pages, the author conveys a good deal of information about first- and second-order ordinary and partial differential equations and their connections to integral equations. The material was developed in classes at the College of William and Mary and at Oxford. The style is somewhat formal, but the topics are handled well, with examples, clear diagrams, and exercises to illuminate and supplement the text. There are no references to using technology and no answers/solutions to exercises appear at the back of the book.
This text assumes a previous course in ordinary differential equations, although some basic material is included in the Appendix. (These days, Chapter 15, on Phase-Plane Analysis, may be considered review material as well.) A very brief sketch of important results in single and multivariable calculus is given in Chapter 0. Linear algebra is introduced where required and an introduction to complex function methods is provided at key places in the text. The author provides different paths through the material via a schematic presentation of the book’s contents and helpful discussions of the interdependence of the chapters. The book ends with a 48-item bibliography.
In contrast to most modern first courses in differential equations, virtually all applications of the material are concerned with classical problems of mathematical physics. For instance, only as an exercise in the last chapter does Volterra’s predator-prey model appear.
As the publisher’s blurb asserts, this is “analysis for applications” and this well-written text is suitable for physicists, economists, chemists, and any student interested in acquiring a working knowledge of applied analysis.
作者在本书中向读者传递了大量的信息，包括一阶与二阶常微分方程、偏微分方程以及它们与积分方程的联系。本书的内容主要整理自威廉玛丽学院与牛津大学微分方程课程的讲义。本书严谨有序，各方面的内容安排得详略得当，配以例子、清晰的图表与习题来对正文部分解释说明。但遗憾的是书后并没有习题的答案。
虽然附录中包含了一些基本内容，但本书仍假定读者修习过常微分方程的入门课程（而关于相平面分析的第15章也可以作为类似附录的对基本内容的复习）。第0章对一元和多元微积分的重要结论作了简要的概述。线性代数在需要时被引入，同时复函数方法也在关键之处被提出。作者通过对书中内容的图解介绍和对各章间关系的讨论，提供了不同的阅读顺序。本书的结尾包含48条参考书目。
与大多数现代微分方程的基础课程不同，几乎所有内容的应用都与数学物理的经典问题有关。例如，Volterra的猎物-捕食者模型作为最后一章的习题出现。
正如出版商的导语所说，这是“应用分析”，本书写得很好，适合物理学家、经济学家、化学家和任何有兴趣对获得应用分析知识感兴趣的学生。
点评人： Henry Ricardo（纽约城市大学梅德加·埃弗斯学院数学教授，MAA大都会NY部门秘书）
Differential and integral equations involve important mathematical techniques, and as such will be encountered by mathematicians, and physical and social scientists, in their undergraduate courses. This text provides a clear, comprehensive guide to first- and second-order ordinary and partial differential equations, whilst introducing important and useful basic material on integral equations. Readers will encounter detailed discussion of the wave, heat and Laplace equations, of Green's functions and their application to the Sturm-Liouville equation, and how to use series solutions, transform methods and phase-plane analysis. The calculus of variations will take them further into the world of applied analysis. Providing a wealth of techniques, but yet satisfying the needs of the pure mathematician, and with numerous carefully worked examples and exercises, the text is ideal for any undergraduate with basic calculus to gain a thorough grounding in 'analysis for applications'.
《Differential and integral equations》囊括重要的数学技巧，并且无论是数学家、物理学家、还是社会学家，都会在本科阶段的课程中遇到书中所提的数学方法。本文清晰又全面地介绍了一二阶常微分与偏微分方程，同时介绍了有关积分方程的重要内容。本书详细讨论了波动方程、热传导方程、Laplace方程、格林函数及其在Sturm-Liouville方程中的应用，以及如何使用幂级数解、变换方法和相平面分析等内容。变分法会将读者们带入更深层次的应用分析世界。提供了丰富的技巧方法，又满足了基础数学家的需要，配以大量的精心设计的例子和练习，本书正适合那些有微积分基础的本科生学习，来为应用分析的学习打下扎实的基础。
资料整理：方诗雨
翻译：赵旭彤