常微分方程基础理论(影印版)/天元基金影印数学丛书(天元基金影印数学丛书)(Basic Theory of Ordinary Differential Equations)

基本信息·出版社：高等教育出版社

·页码：468 页

·出版日期：2007年

·ISBN：9787040220667

·条形码：9787040220667

·包装版本：1版（影印版）

·装帧：其他

·开本：16

·正文语种：英语

·丛书名：天元基金影印数学丛书

·外文书名：Basic Theory of Ordinary Differential Equations

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内容简介本书内容分为四部分：第一部分的内容包括解的存在性、唯一性、对数据的光滑依赖性，以及解的非唯一性；第二部分讨论线性常微分方程；第三部分讨论非线性常微分方程；第四部分讨论常微分方程的幂级数解。

目录

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 ε0

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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