Singular Perturbation Theory

The theory of singular perturbations has evolved as a response to the need to find approximate solutions (in an analytical form) to complex problems. Typically, such problems are expressed in terms of differential equations which contain at least one small parameter, and they can arise in many fields: fluid mechanics, particle physics, and combustion processes, to name but three.

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, is well established. The purpose of the set of volumes to which the present one belongs is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume provides a detailed introduction to a specific subject area of current importance that is summarized below, and then goes beyond this by reviewing recent contributions, and so serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems would otherwise have been impossible.

Written in a form that should enable the relatively inexperienced (or new) worker in the field of singular perturbation theory to learn and apply all the essential ideas Designed as a learning tool. The numerous examples and set exercises are intended to aid this process Includes supplementary material: sn.pub/extras

Texte du rabat

Many areas of science and engineering produce difficult mathematical problems , i.e., problems that cannot be solved in any conventional sense. In many cases, against all the apparent odds, it is possible to construct systematic approximations that lead to useful solutions. The most powerful of these approximation techniques is singular perturbation theory. Singular Perturbation Theory introduces all the background ideas to this subject, designed for those with only the most superficial familiarity with university-level mathematics. The methods are developed through worked examples and set exercises (with answers); the latter part of the book is devoted to applications drawn from: mechanics, physics, semi- and superconductor theory, fluid mechanics, thermal processes, chemical and biochemical reactions. In a novel approach, these are grouped together so that the reader with particular interests can readily access them.

This book is based on material that has been taught, mainly by the author, to MSc and research students in applied mathematics and engineering mathematics at the University of Newcastle upon Tyne over the last thirty years. The aim of this text is to make all the material readily accessible to the reader who wishes to learn and use the ideas to help with research problems and who does not have a strong mathematical background.

Contenu
Mathematical Preliminaries.- Introductory Applications.- Further Applications.- The Method of Multiple Scales.- Some Worked Examples Arising from Physical Problems.