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2007

Product Details

Author Hsieh，P.F and Sibuya，Y

Press Higher Education Press

Isbn 9787040220667

Publication Date July 2007

Pages 468

Size 16K

Edition First

Cover Paperback

Language English

Page material gelatine plate paper

Product Description:

This book is an English photo-engraving one. It is divided to four parts: part one introduces the existence, uniqueness, nonuniqueness and dependence on data; part two talks about linear ordinary differential equation (In this book, it instead Jordan decomposition with The S-N decomposition); part three is mainly about the stability, orbital stability and other geometric theories; part four discusses the solution of power series of ODE.

Content:

Preface

Chapter 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

ChapterⅡ.Dependence on Data

Ⅱ-1.Continuity with respect to initial data and parameters

Ⅱ-2.Differentiability

Exercises Ⅱ

Chapter Ⅲ.Nonuniqueness

Ⅲ-1.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

Chapter 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

Chapter 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.The S-N decomposition of a matrix of infinite order

Ⅴ-4.The S-N decomposition 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

Chapter 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.Refiectionless potentials

Ⅵ-8.Construction of a potential for given data

Ⅵ-9.Differential equations satisfied by reflectionless potentials

Ⅵ-10.Periodic potentials

Exercises Ⅵ

Chapter 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

Chapter 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 R2

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

Chapter 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

Chapter Ⅹ.The Second-Order Differential Equation (d2x)/(dt2)+h(x)*(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 ε > 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

Chapter 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

Chapter 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

Chapter 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

Index

……

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