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Basic Theory of Ordinary Differential Equations
Universitext
ISBN/EAN: 9780387986999
Umbreit-Nr.: 1672073
Sprache:
Englisch
Umfang: xi, 469 S., 114 s/w Illustr., 114 s/w Zeichng.
Format in cm:
Einband:
gebundenes Buch
Erschienen am 22.06.1999
Auflage: 1/1999
- Kurztext
- Inhaltsangabe* Fundamental Theorems of Ordinary Differential Equations * Dependence of Data * Nonuniqueness * General Theory of Linear Systems * 1= Singularities of the First Kind * Boundary-value Problems of Linear Differential Equations of the Second Order * Asymptotic Behavior of Solutions of Linear Systems * Stabiblity * Autonomous Systems * Second Order Differential Equations * Asymptotic Expansions * Asymptotic Expansions in a Parameter * Singularities of the Second Kind 1=
- Autorenportrait
- InhaltsangabeI. 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.- II. Dependence on Data.- II-1. Continuity with respect to initial data and parameters.- II-2. Differentiability.- Exercises II.- III. Nonuniqueness.- III-l. 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.- 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.- 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. TheS-Ndecomposition of a matrix of infinite order.- V-4. TheS-Ndecomposition 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.- 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. Reflectionless potentials.- VI- 8. Construction of a potential for given data.- VI- 9. Differential equations satisfied by reflectionless potentials.- VI-10. Periodic potentials.- Exercises VI.- 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.- 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 ?n.- 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.- IX. Autonomous Systems.- IX-1. Limit-invariant sets.- IX-2. Liapounoff's direct method.- IX-3. Orbital stability.- IX-4. The Poincaré-Bendixson theorem.- IX-5. Indices of Jordan curves.- Exercises IX.- X. The Second-Order Differential Equation $$\frac{{{d^2}x}}{{d{t^2}}} + h(x)\frac{{dx}}{{dt}} + g(x) = 0 $$. X1. Twopoint boundaryvalue problems. X2. Applications of the Liapounoff functions. X3. Existence and uniqueness of periodic orbits. X4. Multipliers of the periodic orbit of the van der Pol equation. X5. The van der Pol equation for a small ?> 0. X6. The van der Pol equation for a large parameter. X7. A theorem due to M. Nagumo. X8. A singular perturbation problem. Exercises X. XI. Asymptotic Expansions. XI1. Asymptotic expansions in the sense of Poincaré. XI2. Gevrey asymptotics. XI3. Flat functions in the Gevrey asymptotics. XI4. Basic properties of Gevrey asymptotic expansions. XI5. Proof of Lemma XI26. Exercises XI. XII. Asymptotic Expansions in a Parameter. XII1. An existence theorem. XII2. Basic estimates. XII3. Proof of Theorem XII12. XII4. A blockdiagonalization theorem. XII5. Gevrey a