Difference between revisions of "Book:Martin Bohner/Dynamic Equations on Time Scales"
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| Line 8: | Line 8: | ||
:Chapter 1. The Time Scales Calculus | :Chapter 1. The Time Scales Calculus | ||
::1.1. Basic Definitions | ::1.1. Basic Definitions | ||
| + | :::[[Time scale|page 1]] | ||
| + | :::[[Forward jump|Definition $1.1$]] | ||
| + | :::[[Induction on time scales|Theorem $1.7$]] | ||
::1.2. Differentiation | ::1.2. Differentiation | ||
| + | :::[[Delta derivative|Definition $1.10$]] | ||
| + | :::[[Delta differentiable implies continuous|Theorem $1.16 (i)$]] | ||
| + | :::[[Delta derivative at right-scattered|Theorem $1.16 (ii)$]] | ||
| + | :::[[Delta derivative at right-dense|Theorem $1.16 (iii)$]] | ||
| + | :::[[Delta simple useful formula|Theorem $1.16 (iv)$]] | ||
| + | :::[[Delta derivative of sum|Theorem $1.20 (i)$]] | ||
| + | :::[[Delta derivative of constant multiple|Theorem $1.20 (ii)$]] | ||
| + | :::[[Delta derivative of product (1)|Theorem $1.20 (iii)$]] (and [[Delta derivative of product (2)|Theorem $1.20 (iii)$]]) | ||
| + | :::[[Delta derivative of reciprocal|Theorem $1.20 (iv)$]] | ||
| + | :::[[Delta derivative of quotient|Theorem $1.20 (v)$]] | ||
| + | :::[[Delta derivative of classical polynomial|Theorem $1.24 (i)$]] | ||
| + | :::[[Delta derivative of reciprocal of classical polynomial|Theorem $1.24 (ii)$]] | ||
::1.3. Examples and Applications | ::1.3. Examples and Applications | ||
::1.4. Integration | ::1.4. Integration | ||
| + | :::[[Regulated function|Definition $1.57$]] | ||
| + | :::[[Rd-continuous|Definition $1.58$]] | ||
| + | :::[[Continuous implies rd-continuous|Theorem $1.60(i)$]] | ||
| + | :::[[Rd-continuous implies regulated|Theorem $1.60(ii)$]] | ||
| + | :::[[Forward jump is rd-continuous|Theorem $1.60(iii)$]] | ||
| + | :::Theorem $1.60(iv)$ | ||
| + | :::Theorem $1.60(v)$ | ||
| + | :::[[Pre-differentiable|Definition $1.62$]] | ||
| + | :::[[Regulated on compact interval is bounded|Theorem $1.65$]] | ||
| + | :::[[Delta mean value theorem|Theorem $1.67$]] | ||
| + | :::Corollary $1.68(i)$ | ||
| + | :::Corollary $1.68(ii)$ | ||
| + | :::Corollary $1.68(iii)$ | ||
::1.5. Chain Rules | ::1.5. Chain Rules | ||
::1.6. Polynomials | ::1.6. Polynomials | ||
| Line 17: | Line 45: | ||
:Chapter 2. First Order Linear Equations | :Chapter 2. First Order Linear Equations | ||
::2.1. Hilger's Complex Plane | ::2.1. Hilger's Complex Plane | ||
| + | :::[[Hilger complex plane]] | ||
| + | :::[[Hilger real axis]] | ||
| + | :::[[Hilger alternating axis]] | ||
| + | :::[[Hilger circle]] | ||
| + | :::[[Hilger real part]] | ||
| + | :::[[Hilger imaginary part]] | ||
| + | :::[[Hilger pure imaginary]] | ||
::2.2. The Exponential Function | ::2.2. The Exponential Function | ||
::2.3. Examples of Exponential Functions | ::2.3. Examples of Exponential Functions | ||
| Line 70: | Line 105: | ||
:Index | :Index | ||
| − | [[Category: | + | [[Category:Book]] |
Latest revision as of 15:27, 15 January 2023
Martin Bohner and Allan Peterson: Dynamic Equations on Time Scales
Online versions
Chapters 1-3 hosted by Martin Bohner
Contents
- Preface
- Chapter 1. The Time Scales Calculus
- 1.1. Basic Definitions
- 1.2. Differentiation
- 1.3. Examples and Applications
- 1.4. Integration
- Definition $1.57$
- Definition $1.58$
- Theorem $1.60(i)$
- Theorem $1.60(ii)$
- Theorem $1.60(iii)$
- Theorem $1.60(iv)$
- Theorem $1.60(v)$
- Definition $1.62$
- Theorem $1.65$
- Theorem $1.67$
- Corollary $1.68(i)$
- Corollary $1.68(ii)$
- Corollary $1.68(iii)$
- 1.5. Chain Rules
- 1.6. Polynomials
- 1.7. Further Basic Results
- 1.8. Notes and References
- Chapter 2. First Order Linear Equations
- 2.1. Hilger's Complex Plane
- 2.2. The Exponential Function
- 2.3. Examples of Exponential Functions
- 2.4. Initial Value Problems
- 2.5. Notes and References
- Chapter 3. Second Order Linear Equations
- 3.1. Wronskians
- 3.2. Hyperbolic and Trigonometric Functions
- 3.3. Reduction of Order
- 3.4. Method of Factoring
- 3.5. Nonconstant Coefficients
- 3.6. Hyperbolic and Trigonometric Functions II
- 3.7. Euler-Cauchy Equations
- 3.8. Variation of Parameters
- 3.9. Annihilator Method
- 3.10. Laplace Transform
- 3.11. Notes and References
- Chapter 4. Self-Adjoint Equations
- 4.1. Preliminaries and Examples
- 4.2. The Riccati Equation
- 4.3. Disconjugacy
- 4.4. Boundary Value Problems and Green's Function
- 4.5. Eigenvalue Problems
- 4.6. Notes and References
- Chapter 5. Linear Systems and Higher Order Equations
- 5.1. Regressive Matrices
- 5.2. Constant Coefficients
- 5.3. Self-Adjoint Matrix Equations
- 5.4. Asymptotic Behavior of Solutions
- 5.5. Higher Order Linear Dynamic Equations
- 5.6. Notes and References
- Chapter 6. Dynamic Inequalities
- 6.1. Gronwall's Inequality
- 6.2. Hölder's and Minkowski's Inequalities
- 6.3. Jensen's Inequality
- 6.4. Opial Inequalities
- 6.5. Lyapunov Inequalities
- 6.6. Upper and Lower Solutions
- 6.7. Notes and References
- Chapter 7. Linear Symplectic Dynamic Systems
- 7.1. Symplectic Systems and Special Cases
- 7.2. Conjoined Bases
- 7.3. Transformation Theory and Trigonometric Systems
- 7.4. Notes and References
- Chapter 8. Extensions
- 8.1. Measure Chains
- 8.2. Nonlinear Theory
- 8.3. Alpha Derivatives
- 8.4. Nabla Derivatives
- 8.5. Notes and References
- Solutions to Selected Problems
- Bibliography
- Index
