圖書信息
中文名: 微分方程數值解
原名: Basic Theory of Ordinary Differential Equations
作者: Po-Fang Hsieh Yasutaka Sibuya 丁同仁 李承治
圖書分類: 科技
版本: 英文版
內容介紹：
《常微分方程基礎理論(影印版)》為開展常微分方程研究工作的讀者提供必要的準備知識，可作為本科高年級和研究生常微分方程課程教材。《常微分方程基礎理論(影印版)》內容分為四部分：第一部分（第一、二、三章）的內容包括解的存在性、唯一性、對數據的光滑依賴性，以及解的非唯一性；第二部分（第四、六、七章）討論線性常微分方程，書中用矩陣的S-N分解代替Jordan分解，前者的計算較後者更容易；第三部分（第八、九、十章）討論非線性常微分方程的穩定性、漸近穩定性等幾何理論；第四部分（第五、十一、十二、十三章）討論常微分方程的冪級數解，包括線性常微分方程的
目錄:
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.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.lytic 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-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
Exercises Ⅻ
Chapter ⅩⅢ.Singularities of the Second Kind
ⅩⅢ xistence 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 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
奇點分類及非線性常微分方程當參數或自變數趨向某奇點時的漸近解等。
