Singular Perturbation Methods for Ordinary Differential Equations

ISBN-13: 9780387975566 / Angielski / Twarda / 1991 / 227 str.

Jr. Robert E. O'Malley

Opis

This book results from various lectures given in recent years. Early drafts were used for several single semester courses on singular perturbation meth- ods given at Rensselaer, and a more complete version was used for a one year course at the Technische Universitat Wien. Some portions have been used for short lecture series at Universidad Central de Venezuela, West Vir- ginia University, the University of Southern California, the University of California at Davis, East China Normal University, the University of Texas at Arlington, Universita di Padova, and the University of New Hampshire, among other places. As a result, I've obtained lots of valuable feedback from students and listeners, for which I am grateful. This writing continues a pattern. Earlier lectures at Bell Laboratories, at the University of Edin- burgh and New York University, and at the Australian National University led to my earlier works (1968, 1974, and 1978). All seem to have been useful for the study of singular perturbations, and I hope the same will be true of this monograph. I've personally learned much from reading and analyzing the works of others, so I would especially encourage readers to treat this book as an introduction to a diverse and exciting literature. The topic coverage selected is personal and reflects my current opin- ions. An attempt has been made to encourage a consistent method of ap- proaching problems, largely through correcting outer limits in regions of rapid change. Formal proofs of correctness are not emphasized.

Spis treści

1: Examples Illustrating Regular and Singular Perturbation Concepts.- A. The Harmonic Oscillator: Low Frequency Situation.- B. Introductory Definitions and Remarks.- C. A Simple First-Order Linear Initial Value Problem.- D. Some Second-Order Two-Point Problems.- E. Regular Perturbation Theory for Initial Value Problems.- 2: Singularly Perturbed Initial Value Problems.- A. A Nonlinear Problem from Enzyme Kinetics.- B. The Solution of Linear Systems Using Transformation Methods.- C. Inner and Outer Solutions of Model Problems.- D. The Nonlinear Vector Problem (Tikhonov—Levinson Theory).- E. An Outline of a Proof of Asymptotic Correctness.- F. Numerical Methods for Stiff Equations.- G. Relaxation Oscillations.- H. A Combustion Model.- I. Linear and Nonlinear Examples of Conditionally Stable Systems.- J. Singular Problems.- 3: Singularly Perturbed Boundary Value Problems.- A. Second Order Linear Equations (without Turning Points).- B. Linear Scalar Equations of Higher Order.- C. First-Order Linear Systems.- D. An Application in Control Theory.- E. Some Linear Turning Point Problems.- F. Quasilinear Second-Order Problems.- G. Existence, Uniqueness, and Numerical Computation of Solutions.- H. Quasilinear Vector Problems.- I. An Example with an Angular Solution.- J. Nonlinear Systems.- K. A Nonlinear Control Problem.- L. Semiconductor Modeling.- M. Shocks and Transition Layers.- N. Phase-Plane Solutions for Conservative Systems.- O. A Geometric Analysis for Some Autonomous Equations.- P. Semilinear Problems.- Appendix: The Historical Development of Singular Perturbations.- References.