Ordinary Differential Equations

ISBN-13: 9780387984599 / Angielski / Twarda / 1998 / 384 str.

Wolfgang Walter; W. Walter; R. Thompson

Opis

Develops the theory of initial-, boundary-, and eigenvalue problems, real and complex linear systems, asymptotic behavior and stability. Using novel approaches to many subjects, the book emphasizes differential inequalities and treats more advanced topics such as Caratheodory theory, nonlinear boundary value problems and radially symmetric elliptic problems. New proofs are given which use concepts and methods from functional analysis. Applications from mechanics, physics, and biology are included, and exercises, which range from routine to demanding, are dispersed throughout the text. Solutions for selected exercises are included at the end of the book. All required material from functional analysis is developed in the book and is accessible to students with a sound knowledge of calculus and familiarity with notions from linear algebra. This text would be an excellent choice for a course for beginning graduate or advanced undergraduate students.

Spis treści

I. First Order Equations: Some Integrable Cases.- § 1. Explicit First Order Equations.- § 2. The Linear Differential Equation. Related Equations.- Supplement: The Generalized Logistic Equation.- § 3. Differential Equations for Families of Curves. Exact Equations.- § 4. Implicit First Order Differential Equations.- II: Theory of First Order Differential Equations.- § 5. Tools from Functional Analysis.- § 6. An Existence and Uniqueness Theorem.- Supplement: Singular Initial Value Problems.- § 7. The Peano Existence Theorem.- Supplement: Methods of Functional Analysis.- § 8. Complex Differential Equations. Power Series Expansions.- § 9. Upper and Lower Solutions. Maximal and Minimal Integrals.- Supplement: The Separatrix.- III: First Order Systems. Equations of Higher Order.- § 10. The Initial Value Problem for a System of First Order.- Supplement I: Differential Inequalities and Invariance.- Supplement II: Differential Equations in the Sense of Carathéodory.- § 11. Initial Value Problems for Equations of Higher Order.- Supplement: Second Order Differential Inequalities.- § 12. Continuous Dependence of Solutions.- Supplement: General Uniqueness and Dependence Theorems.- § 13. Dependence of Solutions on Initial Values and Parameters.- IV: Linear Differential Equations.- § 14. Linear Systems.- § 15. Homogeneous Linear Systems.- § 16. Inhomogeneous Systems.- Supplement:L1-Estimation of C-Solutions.- § 17. Systems with Constant Coefficients.- § 18. Matrix Functions. Inhomogeneous Systems.- Supplement: Floquet Theory.- § 19. Linear Differential Equations of Order n.- § 20. Linear Equations of Order nwith Constant Coefficients.- Supplement: Linear Differential Equations with Periodic Coefficients.- V: Complex Linear Systems.- § 21. Homogeneous Linear Systems in the Regular Case.- § 22. Isolated Singularities.- § 23. Weakly Singular Points. Equations of Fuchsian Type.- § 24. Series Expansion of Solutions.- § 25. Second Order Linear Equations.- VI: Boundary Value and Eigenvalue Problems.- § 26. Boundary Value Problems.- Supplement I: Maximum and Minimum Principles.- Supplement II: Nonlinear Boundary Value Problems.- § 27. The Sturm—Liouville Eigenvalue Problem.- Supplement: Rotation-Symmetric Elliptic Problems.- § 28. Compact Self-Adjoint Operators in Hilbert Space.- VII: Stability and Asymptotic Behavior.- § 29. Stability.- § 30. The Method of Lyapunov.- A. Topology.- B. Real Analysis.- C. C0111plex Analysis.- D. Functional Analysis.- Solutions and Hints for Selected Exercises.- Literature.- Notation.