Asymptotic Expansions for Ordinary Differential Equations

Front Cover
Courier Corporation, 2002 - Mathematics - 374 pages
0 Reviews
"A book of great value . . . it should have a profound influence upon future research."--Mathematical Reviews. Hardcover edition. The foundations of the study of asymptotic series in the theory of differential equations were laid by Poincaré in the late 19th century, but it was not until the middle of this century that it became apparent how essential asymptotic series are to understanding the solutions of ordinary differential equations. Moreover, they have come to be seen as crucial to such areas of applied mathematics as quantum mechanics, viscous flows, elasticity, electromagnetic theory, electronics, and astrophysics. In this outstanding text, the first book devoted exclusively to the subject, the author concentrates on the mathematical ideas underlying the various asymptotic methods; however, asymptotic methods for differential equations are included only if they lead to full, infinite expansions. Unabridged Dover republication of the edition published by Robert E. Krieger Publishing Company, Huntington, N.Y., 1976, a corrected, slightly enlarged reprint of the original edition published by Interscience Publishers, New York, 1965. 12 illustrations. Preface. 2 bibliographies. Appendix. Index.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Some Basic Properties of Linear Differential Equations in the Complex Domain
1
The Basic Existence Theorem and its Consequences
3
Circuit Relations About Singular Points
9
Regular Singular Points
17
Solutions at a Regular Singular Point
20
Asymptotic Power Series
30
Definition of Asymptotic Power Series
31
Elementary Propertles of Asymptotic Series
33
Analytic Simplification
143
Proof of Theorem 26 1
147
Shearing Transformations
151
Turning Point Problems
157
Analytic Theory
169
Short Report on Other Turning Point Problems
185
Nonlinear Equations
197
Solution by Asymptotic Power Series
200

The Existence of Asymptotic Series
39
Irregular Singular Points
49
Formal Simplification
52
Analytic Simplification and Asymptotic Solution
55
Miscellaneous Remarks
61
Proof of the Main Asymptotic Existence Theorem when all Eigenvalues are Distinct
65
The Stokes Phenomenon
76
Generalizations by Means of Jordans Canonical Form
88
General Case
94
General Case
99
General Case
100
Some Special Asymptotic Methods
116
Calculating Asymptotic Expansions from Con
117
vergent Power Series
122
Solution by Laplace Contour Integrals
123
The Saddlepoint Method
127
Asymptotic Expansions with Respect to a Parameter
134
Formal Theory
137
Transformation into a Linear Differential Equation
202
Solution by Exponential Series
213
Nonlinear Equations with a Parameter
217
Singular Perturbations
228
The Method of Visik and Lyusternik
237
Qualitative Theory
249
Series Expansions for the Initial Value Problem
260
Nonlinear TwoPoint Boundary Value Problems
279
Decomposition of General Linear Systems of Singular Perturbation Type
287
General Remarks
299
Linear Theory
305
Series Expansions for Periodic Solutions of Singular Perturbation Problems
315
Integration of Differential Equations by Factorial Series
325
A Brief Summary of Some Recent Research
347
Bibliography
353
Subject Index
369
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page 365 - FWJ Olver, Error bounds for asymptotic expansions, with an application to cylinder functions of large argument, Asymptotic Solutions of Differential Equations and their Applications, edited by CH Wilcox, ( New York, John Wiley and Sons, 1964).
Page 360 - Reduction of the order of a linear ordinary differential equation containing a small parameter,
Page 360 - Asymptotic theory of second order differential equations with two simple turning points,
Page 362 - The solutions of second order ordinary differential equations about a turning point of order two, Trans.
Page 363 - Approximate solution of a system of ordinary differential equations with a small parameter multiplying the derivatives.

References to this book

All Book Search results »

Bibliographic information