Synopses & Reviews
This volume presents an authoritative, unified overview of the methods and results concerning the global properties of linear differential equations of order n (n>=2). It does not, however, seek to be comprehensive. Rather, it contains a selection of results which richly illustrate the unified approach presented. By making use of recent methods and results from many different areas of mathematics and by introducing several original methods, global solutions of problems previously studied only locally are given. The structure of global transformations is described algebraically, and a new geometrical approach is introduced which leads to global canonical forms suitable for Cartan's moving frame-of-reference method. The theory discussed also provides effective tools for solving some open problems, especially relating to the distribution of zeros of solutions. In addition, the theory of functional equations plays an important role in studying the asymptotic behaviour of solutions. Applications to differential geometry and functional equations are also described. The volume is largely self-contained. This book is for mathematicians, computer scientists, physicists, chemists, engineers, and others whose work involves the use of linear differential equations.
Synopsis
This monograph contains a description of original methods and results concern- ing global properties of linear differential equations of the nth order, n 2, in the real domain. This area of research concerning second order linear differential equations was started 35 years ago by O. Boruvka. He summarized his results in the monograph "Lineare Differentialtransforrnationen 2. Ordnung," VEB, Berlin 1967 (extended version: "Linear Differential Transformations of the Second Order," The English U niv. Press, London 1971). This book deals with linear differential equations of the nth order, n 2, and summarizes results in this field in a unified fashion. However, this mono- graph is by no means intended to be a survey of all results in this area. I t contains only a selection of results, which serves to illustrate the unified approach presented here. By using recent methods and results of algebra, topology, differential geometry, functional analysis, theory of functional equations and linear differential equations of the second order, and by introducing several original methods, global solutions of problems which were previously studied only locally by Kummer, Brioschi, Laguerre, Forsyth, Halphen, Lie, Stiickel and others are provided. The structure of global transformations is described by algebraic means (theory of categories: Brandt and Ehresmann groupoids), a new geometrical approach is introduced that leads to global canonical forms (in contrast to the local Laguerre-Forsyth or Halphen forms) and is suitable for the application of Cartan's moving-frame-of-reference method.
Description
Includes bibliographical references (p. [299]-314) and indexes.
Table of Contents
Series Editors Preface. Preface. Symbols defined in the text. 1. Introduction with historical remarks. 2. Notation, definitions and some basic facts. 3. Global transformations. 4. Analytic, algebraic and geometrical aspects of global transformations. 5. Criterion of global equivalence. 6. Stationary groups. 7. Canonical forms. 8. Invariants. 9. Equations with solutions of prescribed properties. 10. Zeros of solutions. 11. Related results and some applications. 12. Appendix: Two functional equations. Literature cited in the book and/or for supplementary reading. Subject index. Index of names.