Ordinary Differential Equations

ISBN-13: 9781461268345 / Angielski / Miękka / 2012 / 384 str.

Wolfgang Walter; R. Thompson

Opis

Based on a translation of the 6th edition of Gewohnliche Differentialgleichungen by Wolfgang Walter, this edition includes additional treatments of important subjects not found in the German text as well as material that is seldom found in textbooks, such as new proofs for basic theorems. This unique feature of the book calls for a closer look at contents and methods with an emphasis on subjects outside the mainstream. Exercises, which range from routine to demanding, are dispersed throughout the text and some include an outline of the solution. Applications from mechanics to mathematical biology are included and solutions of selected exercises are found at the end of the book. It is suitable for mathematics, physics, and computer science graduate students to be used as collateral reading and as a reference source for mathematicians. Readers should have a sound knowledge of infinitesimal calculus and be familiar with basic notions from linear algebra; functional analysis is developed in the text when needed.

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.

O autorze

ORDINARY DIFFERENTIAL EQUATIONS 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. Applications from mechanics, physics, and biology are included, and exercises, which range from routine to demanding, are dispersed throughout the text.