# 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
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
3 Generalized Volterra Integral Equations
3.1 Generalized Volterra Integral Equations of the Second Kind
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
4 Generalized Volterra Integro-Differential Equations
4.1 Generalized Volterra Integro-Differential Equations of the Second Kind
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
5 Generalized Fredholm Integral Equations
5.1 Generalized Fredholm Integral Equations of the Second Kind
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
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