Name: Basic Theory of Ordinary Differential Equations By Po-Fang Hsieh
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Basic Theory of Ordinary Differential Equations Summary

Basic Theory of Ordinary Differential Equations by Po-Fang Hsieh

Providing readers with the very basic knowledge necessary to begin research on differential equations with professional ability, the selection of topics here covers the methods and results that are applicable in a variety of different fields. The book is divided into four parts. The first covers fundamental existence, uniqueness, smoothness with respect to data, and nonuniqueness. The second part describes the basic results concerning linear differential equations, while the third deals with nonlinear equations. In the last part the authors write about the basic results concerning power series solutions. Each chapter begins with a brief discussion of its contents and history, and hints and comments for many problems are given throughout. With 114 illustrations and 206 exercises, the book is suitable for a one-year graduate course, as well as a reference book for research mathematicians.

Basic Theory of Ordinary Differential Equations Reviews

This book in pure mathematics will guide the reader on a newly marked path through classic and awesome terrain---first so well described by Coddington and Levinson---toward research in the important and useful areas of power series solutions and asymptotics. The path not taken would be marked by at least one explicit mention of a PoincarA (c) map, a resonance in celestial mechanics, an average, a traveling wave, or a transversal intersection of separatrices.--MATHEMATICAL REVIEWS

I. Fundamental Theorems of Ordinary Differential Equations.- I-1. Existence and uniqueness with the Lipschitz condition.- I-2. Existence without the Lipschitz condition.- I-3. Some global properties of solutions.- I-4. Analytic differential equations.- Exercises I.- II. Dependence on Data.- II-1. Continuity with respect to initial data and parameters.- II-2. Differentiability.- Exercises II.- III. Nonuniqueness.- III-l. Examples.- III-2. The Kneser theorem.- III-3. Solution curves on the boundary of R(A).- III-4. Maximal and minimal solutions.- III-5. A comparison theorem.- III-6. Sufficient conditions for uniqueness.- Exercises III.- IV. General Theory of Linear Systems.- IV-1. Some basic results concerning matrices.- IV-2. Homogeneous systems of linear differential equations.- IV-3. Homogeneous systems with constant coefficients.- IV-4. Systems with periodic coefficients.- IV-5. Linear Hamiltonian systems with periodic coefficients.- IV-6. Nonhomogeneous equations.- IV-7. Higher-order scalar equations.- Exercises IV.- V. Singularities of the First Kind.- V-1. Formal solutions of an algebraic differential equation.- V-2. Convergence of formal solutions of a system of the first kind.- V-3. TheS-Ndecomposition of a matrix of infinite order.- V-4. TheS-Ndecomposition of a differential operator.- V-5. A normal form of a differential operator.- V-6. Calculation of the normal form of a differential operator.- V-7. Classification of singularities of homogeneous linear systems.- Exercises V.- VI. Boundary-Value Problems of Linear Differential Equations of the Second-Order.- VI- 1. Zeros of solutions.- VI- 2. Sturm-Liouville problems.- VI- 3. Eigenvalue problems.- VI- 4. Eigenfunction expansions.- VI- 5. Jost solutions.- VI- 6. Scattering data.- VI- 7. Reflectionless potentials.- VI- 8. Construction of a potential for given data.- VI- 9. Differential equations satisfied by reflectionless potentials.- VI-10. Periodic potentials.- Exercises VI.- VII. Asymptotic Behavior of Solutions of Linear Systems.- VII-1. Liapounoff's type numbers.- VII-2. Liapounoff's type numbers of a homogeneous linear system.- VII-3. Calculation of Liapounoff's type numbers of solutions.- VII-4. A diagonalization theorem.- VII-5. Systems with asymptotically constant coefficients.- VII-6. An application of the Floquet theorem.- Exercises VII.- VIII. Stability.- VIII- 1. Basic definitions.- VIII- 2. A sufficient condition for asymptotic stability.- VIII- 3. Stable manifolds.- VIII- 4. Analytic structure of stable manifolds.- VIII- 5. Two-dimensional linear systems with constant coefficients.- VIII- 6. Analytic systems in ?n.- VIII- 7. Perturbations of an improper node and a saddle point.- VIII- 8. Perturbations of a proper node.- VIII- 9. Perturbation of a spiral point.- VIII-10. Perturbation of a center.- Exercises VIII.- IX. Autonomous Systems.- IX-1. Limit-invariant sets.- IX-2. Liapounoff's direct method.- IX-3. Orbital stability.- IX-4. The Poincare-Bendixson theorem.- IX-5. Indices of Jordan curves.- Exercises IX.- X. The Second-Order Differential Equation $$\\frac{{{d^2}x}}{{d{t^2}}} + h(x)\\frac{{dx}}{{dt}} + g(x) = 0 $$.- X-1. Two-point boundary-value problems.- X-2. Applications of the Liapounoff functions.- X-3. Existence and uniqueness of periodic orbits.- X-4. Multipliers of the periodic orbit of the van der Pol equation.- X-5. The van der Pol equation for a small ?> 0.- X-6. The van der Pol equation for a large parameter.- X-7. A theorem due to M. Nagumo.- X-8. A singular perturbation problem.- Exercises X.- XI. Asymptotic Expansions.- XI-1. Asymptotic expansions in the sense of Poincare.- XI-2. Gevrey asymptotics.- XI-3. Flat functions in the Gevrey asymptotics.- XI-4. Basic properties of Gevrey asymptotic expansions.- XI-5. Proof of Lemma XI-2-6.- Exercises XI.- XII. Asymptotic Expansions in a Parameter.- XII-1. An existence theorem.- XII-2. Basic estimates.- XII-3. Proof of Theorem XII-1-2.- XII-4. A block-diagonalization theorem.- XII-5. Gevrey asymptotic solutions in a parameter.- XII-6. Analytic simplification in a parameter.- Exercises XII.- XIII. Singularities of the Second Kind.- XIII-1. An existence theorem.- XIII-2. Basic estimates.- XIII-3. Proof of Theorem XIII-1-2.- XIII-4. A block-diagonalization theorem.- XIII-5. Cyclic vectors (A lemma of P. Deligne).- XIII-6. The Hukuhara-Turrittin theorem.- XIII-7. An n-th-order linear differential equation at a singular point of the second kind.- XIII-8. Gevrey property of asymptotic solutions at an irregular singular point.- Exercises XIII.- References.

Additional information

Sku

NLS9781461271710

ISBN 13

9781461271710

ISBN 10

1461271711

Title

Basic Theory of Ordinary Differential Equations by Po-Fang Hsieh

Author

Po-Fang Hsieh

Series

Universitext

Condition

New

Binding type

Paperback

Publisher

Springer-Verlag New York Inc.

Year published

2012-10-24

Number of pages

469

Prizes

N/A

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Book picture is for illustrative purposes only, actual binding, cover or edition may vary.

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This is a new book - be the first to read this copy. With untouched pages and a perfect binding, your brand new copy is ready to be opened for the first time

Basic Theory of Ordinary Differential Equations Po-Fang Hsieh

Basic Theory of Ordinary Differential Equations by Po-Fang Hsieh

Summary

Basic Theory of Ordinary Differential Equations Summary

Basic Theory of Ordinary Differential Equations by Po-Fang Hsieh

Basic Theory of Ordinary Differential Equations Reviews

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