View source for Book:Svetlin G. Georgiev/Integral Equations on Time Scales

{{Book|Integral Equations on Time Scales||||Svetlin G. Georgiev}}

===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
: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
: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
: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
: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

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