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)
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品牌: 赫斯赫
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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