Collatz L. Differential Equations: An Introduction with Applications

John Wiley & Sons, 1986. - 388 pages. ISBN: 0471909556Preface: The purpose of the book is described in the preface to the fifth edition and it has not changed. So only the changes compared with earlier editions are indicated here, without mentioning innumerable smaller additions. In this sixth edition a number of topics have been included, which, though they could not be presented in great detail, have recently become important; it seems to me useful that the reader should at least be acquainted with their formulation. In particular, the sections on partial differential equations have been reconstructed, because these are becoming increasingly important in applications, although, of course, this book cannot be a substitute for the many good treatises on partial differential equations. In this connection I have chosen problems which have proved useful for their applications and for the numerical methods employed; for numerical methods are constantly gaining significance, because as ever more complicated problems are tackled, the prospect of being able to solve a given problem in `closed form' is greatly diminished, and in many cases, particularly in non-linear problems, we have to resort to numerical methods. The concept of (pointwise) monotonicity becomes especially important, because, in cases where it is valid, it presents the only method of obtaining, without excessive labour, usable inclusions of the required solution between upper and lower bounds. Introductions are also given to many problems which have recently become actual, and to methods relating, for example, to free boundary problems, variational equations, incorrectly posed problems, finite-element methods, branching problems, etc.On the question of modernizing the treatment I deemed it necessary to proceed cautiously. There are quite enough very abstract textbooks, often based on functional analysis, about differential equations, in which, however, applications and concrete aspects feature too little. I have preferred much rather that engineers and scientists should be able to understand the treatment. However, in order that access to modern mathematical literature should not be obstructed, I decided to introduce the fundamental existence and uniqueness theorems twice, once in the classical way, and again in the language of functional analysis; the reader will see that both proofs proceed in the same
way.At this point may I be allowed a word of warning: mathematics-which thrives on its applicability-is moving more and more towards a dangerous isolation in that abstractions are overvalued and actualizations are neglected (often even inexcusably). Abstractions were made in order the better to master the concrete, but now mastery of the concrete is in danger of being lost, as may be susbstantiated in innumerable instances. Often a good engineer is better prepared to deal with a differential equation than a mathematician, and mathematics loses ground. This implies a great and increasingly serious danger for mathematics.