Difference between revisions of "Book:Martin Bohner/Dynamic Equations on Time Scales"

From timescalewiki
Jump to: navigation, search
(Created page with "{{Book|Dynamic Equations on Time Scales||||Martin Bohner|author2=Allan Peterson}} ===Online versions=== [http://web.mst.edu/~bohner/sample.pdf Chapters 1-3 hosted by Martin B...")
 
 
(14 intermediate revisions by the same user not shown)
Line 5: Line 5:
  
 
===Contents===
 
===Contents===
 +
:Preface
 +
:Chapter 1. The Time Scales Calculus
 +
::1.1. Basic Definitions
 +
:::[[Time scale|page 1]]
 +
:::[[Forward jump|Definition $1.1$]]
 +
:::[[Induction on time scales|Theorem $1.7$]]
 +
::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
 +
::1.4. 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
 +
::1.6. Polynomials
 +
::1.7. Further Basic Results
 +
::1.8. Notes and References
 +
:Chapter 2. First Order Linear Equations
 +
::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
 +
::2.3. 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
 +
 +
[[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
page 1
Definition $1.1$
Theorem $1.7$
1.2. Differentiation
Definition $1.10$
Theorem $1.16 (i)$
Theorem $1.16 (ii)$
Theorem $1.16 (iii)$
Theorem $1.16 (iv)$
Theorem $1.20 (i)$
Theorem $1.20 (ii)$
Theorem $1.20 (iii)$ (and Theorem $1.20 (iii)$)
Theorem $1.20 (iv)$
Theorem $1.20 (v)$
Theorem $1.24 (i)$
Theorem $1.24 (ii)$
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
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
2.3. 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
Retrieved from "http://timescalewiki.org/index.php?title=Book:Martin_Bohner/Dynamic_Equations_on_Time_Scales&oldid=1963"