Differential-Difference Equations

Name: Differential-Difference Equations
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BELLMAN(Author)

BELLMAN(Editor)

Academic Press

Published on 14. May 2014

483 pages

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978-0-08-095514-8 (ISBN)

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Front Cover
Differential-Difference Equations
Copyright Page
Contents
Dedication
Introduction
Chapter 1. The Laplace Transform
1.1. Introduction
1.2. Existence and Convergence Exercises
1.3. The Inversion Problem
1.4. Behavior of the Dirichlet Kernel
1.5. Analytic Details Exercises
1.6. Statement of Result
1.7. Jump Discontinuity
1.8. Functions of Bounded Variation
1.9. Contour Integration.
1.10. Examples
1.11. The Fejér Transform
1.12. The Inverse Inversion Problem
1.13. The Convolution Theorem Exercises
1.14. The Fourier Transform
1.15. Plancherel-Parseval Theorem
1.16. Application to Laplace Transform
1.17. The Post-Widder Formula
1.18. Real Inversion Formulas Exercise
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 2. Linear Differential Equations
2.1. Introduction
2.2. Linear Differential Equations
2.3. Fundamental Existence and Uniqueness Theorem
2.4. Successive Approximations
2.5. A Fundamental Lemma
2.6. Uniqueness Theorem
2.7. Fixed-point Techniques
2.8. Difference Schemes
2.9. The Matrix Equation
2.10. Alternative Derivation
2.11. The Inhomogeneous Equation
2.12. The Adjoint Equation
2.13. Constant Coefficients-I
2.14. Constant Coefficients-II
2.15. Laplace Transform Solution
2.16. Characteristic Values and Characteristic Funotions
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 3. First-order Linear Differential-Difference Equations of Retarded Type with Constant Coefficients
3.1. Introduction
3.2. Examples Exercises
3.3. Equations of Retarded, Neutral, and Advanced Type
3.4. The Existence-Uniqueness Theorem Exercises
3.5. Exponential Solutions Exercises
3.6. Order of Growth of Solutions Exercises
3.7. Laplace Transform Solution Exercises
3.8. Solution of a Differential Equation in the Form of a Definite Integral
3.9. Solution of a Differential-Difference Equation in the Form of a Definite Integral Exercises
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 4. Series Expansions of Solutions of First-order Equations of Retarded Type
4.1. The Characteristic Roots
4.2. Series Expansions
4.3. Other Forms of the Expansion Theorem Exercises
4.4. Asymptotic Behavior of the Solution Exercises
4.5. Stability of Equilibrium Exercises
4.6. Fourier-type Expansions Exercises
4.7. The Shift Theorem Exercises
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 5. First-order Linear Equations of Neutral and Advanced Type with Constant Coefficients
5.1. Existence-Uniqueness Theorems Exercises
5.2. Solution by Exponentials and by Definite Integrals: Equations of Neutral Type Exercises
5.3. Series Expansions: Equations of Neutral Type Exercise
5.4. Asymptotic Behavior and Stability: Equations of Neutral Type
5.5. Other Expansions for Solutions of Equations of Neutral Type Exercises
5.6. Equations of Advanced Type
Miscellaneous Exercises and Research Problems
Chapter 6. Linear Systems of Differential-Difference Equations with Constant Coefficients
6.1. Introduction
6.2. Vector-matrix Notation
6.3. Classification of Systems
6.4. Existence-Uniqueness Theorems for Systems Exercises
6.5. Transform Solutions: Retarded-Neutral Systems Exercises
6.6. Solution of Neutral and Retarded Systems by Definite Integrals Exercises
6.7. Series Expansions for Neutral and Retarded Systems Exercises
6.8. Asymptotic Behavior of Solutions of Neutral and Retarded Systems Exercises
6.9. Scalar Equations Exercises
6.10. The Finite Transform Method Exercise
6.11. Fourier-type Expansions Exercises
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 7. The Renewal Equation
7.1. Introduction
7.2. Existence and Uniqueness Exercises
7.3. Further Existence and Uniqueness Theorems
7.4. Monotonicity and Bounded Variation Exercises
7.5. The Formal Laplace Transform Solution
7.6. Exponential Bounds for u(t)
7.7. Rigorous Solution
7.8. A Convolution Theorem
7.9. Asymptotic Behavior of Solutions
7.10. Use of the Contour Integral Representation
7.11. ø(t) a Positive Function
7.12. Shift of the Contour
7.13. Step Functions
7.14. An Elementary Result
7.15. A Less Easily Obtained Result
7.16. Abelian and Tauberian Results
7.17. A Tauberian Theorem of Hardy and Littlewood
7.18. Asymptotic Behavior of Solution of Renewal Equation
7.19. Discussion
7.20. A Tauberian Theorem of Ikehara
7.21. The Tauberian Theorem of Wiener
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 8. Systems of Renewal Equations
8.1. Introduction
8.2. Vector Renewal Equation
8.3. Positive Matrices
8.4. Some Consequences
8.5. Zero with Largest Real Part
8.6. Asymptotic Behavior
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 9. Asymptotic Behavior of linear Differential-Difference Equations
9.1. Introduction
9.2. First Principal Result
9.3. Preliminaries
9.4. Discussion
9.5. ?8 |a(t1)|dt1 &8
9.6. The Difficult Part of Theorem 9.1
9.7. A Lemma
9.8. Continuation of Proof of Theorem 9.1
9.9. The Case Where b(t) ? 0
9.10. Further Results
9.11. More Precise Results
9.12. Asymptotic Series
9.13. The Foundations of Asymptotic Series
9.14. Alternative Formulation
9.15. Differential and Integral Properties
9.16. Extension of Definition Exercises
9.17. First-order Linear Differential Equations
9.18. Second-order Linear Differential Equations
9.19. The Case Where a0 = 0 Exercise
9.20. A Rigorous Derivation of the Asymptotic Expansion
9.21. Determination of the Constants Exercises
9.22. A Basic Problem in the Theory of Differential Equations
9.23. Formal Determination of Coefficients
9.24. Asymptotic Expansion of Solution
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 10. Stability of Solutions of linear Differential-Difference Equations
10.1. Introduction.
10.2. Stability Theory for Ordinary Differential Equations
10.3. The Adjoint Equation
10.4. The Scalar Linear Differential-Difference Equation
10.5. The Matrix Equation with Retarded Argument
10.6. AStability Theorem for Equations with Retarded Argument
10.7. Equations with Constant Coefficients
10.8. A Lemma
10.9. A Stability Theorem for Equations with Constant Coefficients
10.10. Boundedness of Solutions of the Unperturbed System
10.11. The Scalar Equation of Neutral Type: Integral Representation for a Solution Exercise
10.12. The Scalar Equation of Neutral Type: Representation for the Derivative of a Solution
10.13. Systems of Equations of Neutral Type
10.14. Stability Theorems for Equations of Neutral Type
10.15. Stability Theorems for Equations of Neutral Type with Constant Coefficients
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 11. Stability Theory and Asymptotic Behavior for Nonlinear Differential-Difference Equations
11.1. Introduction
11.2. The Poincaré-Liapunov Theorem
11.3. Small Perturbations for General Systems
11.4. Types of Stability
11.5. Existence Theorem for Nonlinear Differential-Difference Equations
11.6. Uniqueness
11.7. Statement of Existence and Uniqueness Theorems Exercises
11.8. Stability Theorem Exercise
11.9. Stability Theorem: Second Proof Exercise
11.10. Asymptotic Behavior of the Solutions.
11.11. Proof of Theorem 11.5 Exercises
11.12. Another Stability Theorem
11.13. Dini-Hukuhara Theorem for Equations with Variable Coefficients
11.14. Poincaré-Liapunov Theorem for Equations with General Variable Coefficients
11.15. Asymptotic Behavior for Nonlinear Equations with Almost-constant Coefficients
11.16. Systems of Nonlinear Equations
11.17. Liapunov Functions and Functionals
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 12. Asymptotic Location of the Zeros of Exponential Polynomials
12.1. Introduction
12.2. The Form of det H(s)
12.3. Zeros of Analytic Functions Exercises
12.4. Constant Coefficients and Commensurable Exponents
12.5. Constant Coefficients and General Real Exponents
12.6. Asymptotically Constant Coefficients
12.7. Polynomial Coefficients with mj and ßj Proportional
12.8. Polynomial Coefficients
12.9. Examples Exercise
12.10. Conditions That All Roots Be of Specified Type Exercises
12.11. Construction of Contours
12.12. Order Results for H-1(S)
12.13. Order Results in the Scalar Case
12.14. Convergence of Integrals over the Contours Exercises
12.15. Integrals along Vertical Lines
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Chapter 13. On Stability Properties of the Zeros of Exponential Polynomials
13.1. Introduction
13.2. Exponential Polynomials Exercises
13.3. Functions of the Form f(z, cos z, sin z)
13.4. Presence of a Principal Term
13.5. Zeros of h(z, ez )
13.6. The Fundamental Stability Results
13.7. A Result of Hayes
13.8. An Important Equation
13.9. Another Example
Miscellaneous Exercises and Research Problems
Bibliography and Comments
Author Index
Subject Index
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