Meyer / Parter Singular Perturbations and Asymptotics

Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, May 28-30, 1980
1. Auflage 2014
ISBN: 978-1-4832-6457-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 418 Seiten, Web PDF

Mathematics Research Center Symposia and Advanced Seminar Series: Singular Perturbations and Asymptotics covers the lectures presented at an Advanced Seminar on Singular Perturbation and Asymptotics, held in Madison, Wisconsin on May 28-30, 1980 under the auspices of the Mathematics Research Center of the University of Wisconsin-Madison. The book focuses on the processes, methodologies, reactions, and principles involved in singular perturbations and asymptotics, including boundary value problems, equations, perturbations, water waves, and gas dynamics. The selection first elaborates on basic concepts in the analysis of singular perturbations, limit process expansions and approximate equations, and results on singularly perturbed boundary value problems. Discussions focus on quasi-linear and nonlinear problems, semilinear systems, water waves, expansion in gas dynamics, asymptotic matching principles, and classical perturbation analysis. The text then takes a look at multiple solutions of singularly perturbed systems in the conditionally stable case and singular perturbations, stochastic differential equations, and applications. The book ponders on connection problems in the parameterless case; general connection-formula problem for linear differential equations of the second order; and turning-point problems for ordinary differential equations of hydrodynamic type. Topics include the comparison equation method, boundary layer flows, compound matrix method, asymptotic solution of the connection-formula problem, and higher order equations. The selection is a valuable source of information for researchers interested in singular perturbations and asymptotics.

Autoren/Hrsg.

Meyer, Richard E.

Parter, Seymour V.

Inhaltsverzeichnis

1;Front Cover;1
2;Singular Perturbations and Asymptotics;4
3;Copyright Page;5
4;Table of Contents;6
5;Contributors;8
6;Preface;10
7;Part I: Theory of Singular Layer Problems;12
7.1;Chapter 1. On Some Basic Concepts in the Analysis of Singular Perturbations;12
7.1.1;1. CLASSICAL PERTURBATION ANALYSIS;13
7.1.2;2. THE GENERAL PROCEDURE IN SINGULAR PERTURBATIONS;15
7.1.3;3. REGULAR EXPANSIONS AND LOCAL EXPANSIONS;18
7.1.4;4. THE CORRESPONDENCE PRINCIPLE;19
7.1.5;5. MATCHING OF EXPANSIONS. EXTENSION THEOREM AND THE OVERLAP HYPOTHESIS;22
7.1.6;6 . ASYMPTOTIC MATCHING PRINCIPLES;25
7.1.7;7. SOME MONOGRAPHS ON SINGULAR PERTURBATIONS;27
7.2;Chapter 2. Limit Process Expansions and Approximate Equations;30
7.2.1;1. INTRODUCTION;30
7.2.2;2. EXPANSIONS IN GAS DYNAMICS;30
7.2.3;3. WATER WAVES;46
7.2.4;4. REMARKS;50
7.3;Chapter 3. Some Old and New Results on Singularly Perturbed Boundary Value Problems;52
7.3.1;1. INTRODUCTION;52
7.3.2;2. SEMI LINEAR PROBLEMS;53
7.3.3;3. QUASILINEAR PROBLEMS;60
7.3.4;4. QUADRATIC PROBLEMS;70
7.3.5;5. SUPERQUADRATIC PROBLEMS;76
7.3.6;6. NOTES AND COMMENTS;79
7.3.7;7. SEMILINEAR PROBLEMS;80
7.3.8;8. NONLINEAR PROBLEMS;82
7.3.9;9. NOTES AND COMMENTS;86
7.3.10;10. SEMILINEAR SYSTEMS;87
7.3.11;11. QUASILINEAR PROBLEMS;88
7.3.12;12. NOTES AND COMMENTS;90
7.3.13;13. CONCLUDING REMARKS;90
7.3.14;ACKNOWLEDGEMENTS;90
7.3.15;REFERENCES;91
7.4;Chapter 4. On Multiple Solutions of Singularly Perturbed Systems in the Conditionally Stable Case;98
7.4.1;1. INTRODUCTION;98
7.4.2;2. THE ASYMPTOTIC APPROXIMATIONS;99
7.4.3;3. A SIMPLE EXAMPLE;108
7.4.4;APPENDIX: THE CONSTRUCTION OF IMPULSIVE SOLUTIONS TO QUASILINEAR PROBLEMS;112
7.4.5;REFERENCES;115
7.4.6;ACKNOWLEDGEMENTS;118
8;Part II: II. Resonance in Singular Perturbations and Applications to Physical Chemistry;120
8.1;Chapter 5. Singular Perturbations, Stochastic Differential Equations, and Applications;120
8.1.1;1. INTRODUCTION;120
8.1.2;2. SINGULAR PERTURBATIONS;125
8.1.3;3. APPLICATIONS;136
8.1.4;REFERENCES;156
8.2;Chapter 6. The Singularly Perturbed Turning-Point Problem: A Spectral Approach;160
8.2.1;1. INTRODUCTION;160
8.2.2;2. MOTIVATION BY AN EXAMPLE;162
8.2.3;3. FIRST-ORDER APPROXIMATIONS OF THE EIGENVALUES;163
8.2.4;4. APPROXIMATIONS OF HIGHER ORDER TO THE EIGENVALUES AND EIGENFUNCTIONS;166
8.2.5;5. EXPONENTIAL DECAY AND RESONANCE (THE CASE p ' (0) > 0);170
8.2.6;6. A MULTIPLE TURNING POINT PROBLEM;172
8.2.7;REFERENCES;182
8.3;Chapter 7. The Singularly Perturbed Turning-Point Problem: A Geometric Approach;184
8.3.1;1. INTRODUCTION;184
8.3.2;2. DEFINITION OF RESONANCE;186
8.3.3;3. NECESSARY CONDITIONS;186
8.3.4;4. THE SIMPLEST EXAMPLE;187
8.3.5;5. THE GENERAL CASE;189
8.3.6;6. CONSTRUCTING REALIZATIONS OF OUTER EXPANSIONS;193
8.3.7;7. THE BOUNDARY VALUE PROBLEM;195
8.3.8;8. COMPUTATIONS USING ASYMPTOTIC EXPANSIONS;198
8.3.9;REFERENCES;200
9;Part III: Multivariate Methods and Applications;202
9.1;Chapter 8. Passage through Resonance;202
9.1.1;1. INTRODUCTION AND HISTORICAL BACKGROUND;202
9.1.2;2. NEARLY PERIODIC HAMILTONIAN SYSTEMS;205
9.1.3;3. IRREDUCIBLE SYSTEMS OF HIGHER ORDER;215
9.1.4;REFERENCES;231
9.2;Chapter 9. A Comparison of Perturbation Methods for Nonlinear Hyperbolic Waves;234
9.2.1;1. INTRODUCTION;234
9.2.2;2. SIMPLE WAVES;237
9.2.3;3. OPPOSITELY-TRAVELING WAVES;250
9.2.4;4. EFFECT OF DISSIPATION;257
9.2.5;5. WAVES INDUCED BY AN INFINITE PLATE;261
9.2.6;6. DIRECTIONAL CYLINDRICAL WAVES;271
9.2.7;7. DIRECTIONAL SPERICAL WAVES;276
9.2.8;8. CONCLUDING REMARKS;280
9.2.9;REFERENCES;281
9.2.10;ACKNOWLEDGMENT;287
9.3;Chapter 10. Asymptotic Calculus of Variations;288
9.3.1;INTRODUCTION;288
9.3.2;1. PERIODIC STRUCTURES AND AVERAGING PRINCIPLE;290
9.3.3;2. FLOWS IN MEDIA WITH PERIODIC OBSTACLES;295
9.3.4;3. ASYMPTOTIC OPTIMAL CONTROL. ELLIPTIC STATE EQUATION;298
9.3.5;4. OTHER EXAMPLES ARISING IN OPTIMAL CONTROL;302
9.3.6;REFERENCES;306
10;Part IV: Turning-Point Theory and Applications;308
10.1;Chapter 11. Turning-Point Problems for Ordinary Differential Equations of Hydrodynamic Type;308
10.1.1;1. INTRODUCTION;308
10.1.2;2. THE HEURISTIC THEORY;309
10.1.3;3. THE GENERALIZED AIRY FUNCTIONS;311
10.1.4;4. A PRELIMINARY TRANSFORMATION;313
10.1.5;5. THE COMPARISON EQUATION METHOD;314
10.1.6;6. THE GENERALIZED AIRY FUNCTION METHOD;316
10.1.7;7. THE EIGENVALUE RELATION;319
10.1.8;8. THE COMPOUND MATRIX METHOD;320
10.1.9;9. BOUNDARY LAYER FLOWS;322
10.1.10;10. STRATIFIED VISCOUS SHEAR FLOWS;323
10.1.11;REFERENCES;325
10.2;Chapter 12. The General Connection-Formula Problem for Linear Differential Equations of the Second Order;328
10.2.1;ABSTRACT;328
10.2.2;1. INTRODUCTION;329
10.2.3;2. THE CONNECTION-FORMULA PROBLEM;333
10.2.4;3. THE LIOUVILLE-GREEN APPROXIMATION THEOREM;339
10.2.5;4. ASYMPTOTIC SOLUTION OF THE CONNECTION-FORMULA PROBLEM;343
10.2.6;5. EXAMPLE;347
10.2.7;6. CONCLUSIONS;351
10.2.8;REFERENCES;352
10.3;Chapter 13. Connection Formulas and Behavior in the Large for Solutions of Linear Differential Equations Depending Singularly on a Parameter;356
10.3.1;1. INTRODUCTION;356
10.3.2;2. CANONICAL REGIONS AND CONNECTIONS AROUND TURNING POINTS;357
10.3.3;3. HIGHER ORDER EQUATIONS;360
10.3.4;4. APPLICATIONS;362
10.3.5;REFERENCES;363
10.4;Chapter 14. Connection Problems in the Parameterless Case: Progress and More Problems;368
10.4.1;1. INTRODUCTION;368
10.4.2;2. CONNECTION PROBLEMS;369
10.4.3;3. REDUCTION TO A PROBLEM OF DIFFERENCE EQUATIONS; THE METHOD OF OKUBO AND KOHNO;374
10.4.4;4. REDUCTION TO REGULAR SINGULAR CONNECTION PROBLEMS: METHODS OF R. SCHAFKE AND BALSER, JURKAT, AND LUTZ;379
10.4.5;5. FURTHER PROBLEMS;385
10.4.6;REFERENCES;387
10.5;Chapter 15. Some New Results on Power-Series Solutions of Algebraic Differential Equations;390
10.5.1;I. SOME WELL-KNOWN RESULTS;390
10.5.2;II. NEW RESULTS;403
10.5.3;REFERENCES;414
11;Index;416