Singular Perturbation Theory

by Johnson, R. S.

• ISBN13: 9780387232003
• ISBN10: 0387232001
• Format: Hardcover
• Copyright: 2004-12-31
• Publisher: Springer Verlag

Summary

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.

Table of Contents

Foreword	xi
Preface	xiii
Mathematical preliminaries	1	(46)
Some introductory examples	2	(8)
Notation	10	(3)
Asymptotic sequences and asymptotic expansions	13	(3)
Convergent series versus divergent series	16	(4)
Asymptotic expansions with a parameter	20	(2)
Uniformity or breakdown	22	(4)
Intermediate variables and the overlap region	26	(2)
The matching principle	28	(4)
Matching with logarithmic terms	32	(3)
Composite expansions	35	(12)
Further Reading	40	(1)
Exercises	41	(6)
Introductory applications	47	(68)
Roots of equations	47	(8)
Integration of functions represented by asymptotic expansions	55	(4)
Ordinary differential equations: regular problems	59	(7)
Ordinary differential equations: simple singular problems	66	(9)
Scaling of differential equations	75	(5)
Equations which exhibit a boundary-layer behaviour	80	(6)
Where is the boundary layer?	86	(4)
Boundary layers and transition layers	90	(25)
Further Reading	103	(1)
Exercises	104	(11)
Further applications	115	(42)
A regular problem	116	(2)
Singular problems I	118	(10)
Singular problems II	128	(11)
Further applications to ordinary differential equations	139	(18)
Further Reading	147	(1)
Exercises	148	(9)
The method of multiple scales	157	(40)
Nearly linear oscillations	157	(8)
Nonlinear oscillators	165	(3)
Applications to classical ordinary differential equations	168	(8)
Applications to partial differential equations	176	(7)
A limitation on the use of the method of multiple scales	183	(1)
Boundary-layer problems	184	(13)
Further Reading	188	(1)
Exercises	188	(9)
Some worked examples arising from physical problems	197	(72)
Mechanical & electrical systems	198	(21)
Celestial mechanics	219	(7)
Physics of particles and of light	226	(9)
Semi- and superconductors	235	(7)
Fluid mechanics	242	(13)
Extreme thermal processes	255	(7)
Chemical and biochemical reactions	262	(7)
Appendix: The Jacobian Elliptic Functions	269	(2)
Answers and Hints	271	(12)
References	283	(4)
Subject Index	287

Excerpts

"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."--BOOK JACKET.