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  • 常微分方程基础理论(影印版)[平装]
  • 共1个商家     29.20元~29.20
  • 作者:赫斯赫(作者),赛拜雅(作者)
  • 出版社:高等教育出版社;第1版(2007年7月1日)
  • 出版时间:
  • 版次 :
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  • 包装:
  • ISBN:9787040220667

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    《常微分方程基础理论(影印版)》的引进是为了更好地借鉴国外微积分教学与研究的成功经验,促进我国数学教育与研究事业的发展,提高高等学校数学教育教学质量,为本科高年级和研究生开展常微分程研究工作提供必要的理论依据,《常微分方程基础理论(影印版)》为原版影印,既可供本科高年级和研究生自学参考,也可做为教材使用。

    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 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

    序言

    为了更好地借鉴国外数学教育与研究的成功经验,促进我国数学教育与研究事业的发展,提高高等学校数学教育教学质量,本着“为我国热爱数学的青年创造一个较好的学习数学的环境”这一宗旨,天元基金赞助出版“天元基金影印数学丛书”。
    该丛书主要包含国外反映近代数学发展的纯数学与应用数学方面的优秀书籍,天元基金邀请国内各个方向的知名数学家参与选题的工作,经专家遴选、推荐,由高等教育出版社影印出版。为了提高我国数学研究生教学的水平,暂把选书的目标确定在研究生教材上。当然,有的书也可作为高年级本科生教材或参考书,有的书则介于研究生教材与专著之间。
    欢迎各方专家、读者对本丛书的选题、印刷、销售等工作提出批评和建议。

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