# Book:Svetlin G. Georgiev/Integral Equations on Time Scales

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## Svetlin G. Georgiev: *Integral Equations on Time Scales*

### Contents

- 1 Elements of the Time Scale Calculus
- 1.1 Forward and Backward Jump Operators, Graininess Function
- 1.2 Differentiation
- 1.3 Mean Value Theorems
- 1.4 Integration
- 1.5 The Exponential Function
- 1.5.1 Hilger's Complex Plane
- 1.5.2 Definition and Properties of the Exponential Function
- 1.5.3 Examples for Exponential Functions

- 1.6 Hyperbolic and Trigonometric Functions
- 1.7 Dynamic Equations
- 1.8 Advanced Practical Exercises

- 2 Introductory Concepts of Integral Equations on Time Scales
- 2.1 Reducing Double Integrals to Single Integrals
- 2.2 Converting IVP to Generalized Volterra Integral Equations
- 2.3 Converting Generalized Volterra Integral Equations to IVP
- 2.4 Converting BVP to Generalized Fredholm Integral Equation
- 2.5 Converting Generalized Fredholm Integral Equation to BVP
- 2.6 Solutions of Generalized Integral Equations and Generalized Integro-Differential Equations
- 2.7 Advanced Practical Exercises

- 3 Generalized Volterra Integral Equations
- 3.1 Generalized Volterra Integral Equations of the Second Kind
- 3.1.1 The Adomian Decomposition Method
- 3.1.2 The Modified Decomposition Method
- 3.1.3 The Noise Terms Phenomenon
- 3.1.4 Differential Equations Method
- 3.1.5 The Successive Approximations Method

- 3.2 Conversion of a Generalized Volterra Integral Equation of the First Kind to a Generalized Volterra Integral Equation of the Second Kind
- 3.3 Existence and Uniqueness of Solutions
- 3.3.1 Preliminary Results
- 3.3.2 Existence of Solutions of Generalized Volterra Integral Equations of the Second Kind
- 3.3.3 Uniqueness of Solutions of Generalized Volterra Integral Equations of the Second Kind
- 3.3.4 Existence and Uniqueness of Solutions of Generalized Volterra Integral Equations of the First Kind

- 3.4 Resolvent Kernels
- 3.5 Application to Linear Dynamic Equations
- 3.6 Advanced Practical Exercises

- 3.1 Generalized Volterra Integral Equations of the Second Kind
- 4 Generalized Volterra Integro-Differential Equations
- 4.1 Generalized Volterra Integro-Differential Equations of the Second Kind
- 4.1.1 The Adomian Decomposition Method
- 4.1.2 Converting Generalized Volterra Integro-Differential Equations of the Second Kind to Initial Value Problems
- 4.1.3 Converting Generalized Volterra Integro-Differential Equations of the Second Kind to Generalized Volterra Integral Equations

- 4.2 Generalized Volterra Integro-Differential Equations of the First Kind
- 4.3 Advanced Practical Exercises

- 4.1 Generalized Volterra Integro-Differential Equations of the Second Kind
- 5 Generalized Fredholm Integral Equations
- 5.1 Generalized Fredholm Integral Equations of the Second Kind
- 5.1.1 The Adomian Decomposition Method
- 5.1.2 The Modified Decomposition Method
- 5.1.3 The Noise Terms Phenomenon
- 5.1.4 The Direct Computation Method
- 5.1.5 The Successive Approximations Method

- 5.2 homogeneous Generalized Fredholm Integral Equations of the Second Kind
- 5.3 Fredholm Alternative Theorem
- 5.3.1 The Case When $\displaystyle\int_a^b \displaystyle\int_a^b |K(X,Y)|^2 \Delta X \Delta Y < 1$
- 5.3.2 The General Case
- 5.3.3 Fredholm's Alternative Theorem

- 5.4 The Schmidt Expansion Theorem and the Mercer Expansion Theorem
- 5.4.1 Operator-Theoretical Notations
- 5.4.2 The Schmidt Expansion Theorem
- 5.4.3 Application to Generalized Fredholm Integral EQuation of the First Kind
- 5.4.4 Positive Definite Kernels. Mercer's Expansion Theorem

- 5.5 Advanced Practical Exercises

- 5.1 Generalized Fredholm Integral Equations of the Second Kind
- 6 Hilbert-Schmidt Theory of Generalized Integral Equations with Symmetric Kernels
- 6.1 Schmidt's Orthogonalization Process
- 6.2 Approximations of Eigenvalues
- 6.3 Inhomogeneous Generalized Integral Equations

- 7 The Laplace Transform Method
- 7.1 The Laplace Transform
- 7.1.1 Definition and Examples
- 7.1.2 Properties of the Laplace Transform
- 7.1.3 Convolution and Shifting Properties of Special Functions

- 7.2 Applications to Dynamic Equations
- 7.3 Generalized Voltera Integral Equations of the Second Kind
- 7.4 Generalized Voltera Integral Equations of the First Kind
- 7.5 Generalized Volterra Integro-Differential Equations of the Second Kind
- 7.6 Generalized Volterra Integro-Differential Equations of the First Kind
- 7.7 Advanced Practical Exercises

- 7.1 The Laplace Transform
- 8 The Series Solution Method
- 8.1 Generalized Volterra Integral Equations of the Second Kind
- 8.2 Generalized Volterra Integral Equations of the First Kind
- 8.3 Generalized Volterra Integro-Differential Equations of the Second Kind

- 9 Non-linear Generalized Integral Equations
- 9.1 Non-linear Generalized Volterra Integral Equations
- 9.2 Non-linear Generalized Fredholm Integral Equations

- References
- Index