Understanding Integro differential Equations 1st Edition by Vasundhara Devi, Zahia Drici, Farzana McRae ISBN 9798891130401 8891130400

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ISBN 10: 8891130400
ISBN 13: 9798891130401
Author: Vasundhara Devi, Zahia Drici, Farzana McRae 

Understanding Integro differential Equations 1st Edition Table of contents:

Chapter 1AN OVERVIEW OF EXISTENCE RESULTS FORSOME INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

1. Introduction

2. Integral Equations

3. Classification of Integral Equations

3.1. Fredholm Integral Equation

3.1.1. Fredholm Integral Equation of First Kind

3.1.2. Fredholm Integral Equation of Second Kind

3.2. Volterra Integral Equation

3.2.1. Volterra Integral Equation of First Kind

3.2.2. Volterra Integral Equation of Second Kind

4. Integro-Differential Equations

4.1. Classification of Integro-Differential Equations

4.1.1. Fredholm Integro-Differential Equation

4.1.2. Fredholm Integro-Differential Equation of Second Kind

4.1.3. Volterra Integro-Differential Equation

4.1.4. Volterra-Fredhlom Integro-Differential Equation

5. Existence Using Fixed Point Theorems

5.1. Existence Result Using Tychonoff Fixed Point Theorem

5.2. Existence Using Schauder Fixed Point Theorem

6. Monotone Iterative Technique

6.1. InDE with Retardation and Anticipation

6.2. PBVP of InDE

6.3. MIT for Higher Order Integro-Differential Equations

7. Quasilinearization

Conclusion

References

Chapter 2 A BRIEF OVERVIEW OF THE STABILITY THEORY OF INTEGRO-DIFFERENTIAL EQUATIONS

Abstract

1. Introduction

2. Part I – Linear Integro-Differential Equations

2.1. InDE of Convolution Type

2.2. InDE of Convolution Type with Infinite Memory

2.3. Integral as Perturbation

2.4. Linear InDE with Nonlinear Perturbation

2.5. Quasilinear InDS

2.6. Linear Barbashin Type InDE

2.7. Impulsive InDS

3. Part II – Nonlinear Integro-Differential Equations

3.1. Introduction

3.2. Lyapunov Functionals

3.2.1. Existence of Lyapunov Functionals

3.2.2. Lyapunov Functional for a Specific Scalar InDE

3.2.3. Lyapunov Functional for a General Scalar InDE

3.3. Method of Lyapunov Functions

3.3.1. Stability Using Minimal Classes

3.4. Lyapunov Functions on Product Spaces

3.4.1. Stability in Two Measures

3.4.2. Practical Stability for InDE

Conclusion

References

Chapter 3Advances in the Qualitative Theoryof Integro–Differential Equations

Abstract

1. Introduction

2. Basic Results

3. Lyapunov and Lyapunov –Krasovskii QualitativeResults of IDEs

4. Ulam Stability, Ulam- Hyers Stability, Ulam -Hyers-RassiasStability of IDEs

Conclusion

References

Chapter 4MATHEMATICAL METHODS FORINTEGRO-DIFFERENTIAL EQUATIONS ANDTHEIR APPLICATIONS

Abstract

1. Introduction

2. Differential TransformMethod

2.1. Fredholm Integro-Differential Equations

One-dimensional differential transform

Two-dimensional differential transform

2.2. Higher-Order Integro-Differential Equations

2.3. Numerical Approach of Differential Transform Method

3. Numerical Results

4. The Adomain DecompositionMethod

5. The Laplace DecompositionMethod

5.1. Laplace Transform Method

6. The Variational IterationMethod

Conclusion

References

Chapter 5OPERATIONAL MATRICES FOR SOLVINGFRACTIONAL ORDER INTEGRAL ANDINTEGRO-DIFFERENTIAL EQUATIONS

Abstract

1. Introduction

2. Fractional Calculus

2.1. Riemann-Liouville Fractional Integral and Derivative

2.2. Caputo Fractional Derivative

3. Legendre Polynomials and Related Results

4. Approximation of Functions in Terms of LegendrePolynomials

5. Operational Matrices Based on Legendre Polynomi-als

5.1. The Operational Matrix of Riemann-Liouville Fractional In-tegral

5.2. The OperationalMatrix of the Product

6. Applications

6.1. Solving Fractional Differential Equation

6.2. Solving Fractional Integro-Differential Equation

7. Existence, Uniqueness, Stability and ConvergenceResults

7.1. Existence and Uniqueness

7.2. Ulam-Hyers Stability

7.3. Convergence Analysis

Conclusion

References

Chapter 6 ON A SECOND ORDER NONLINEAR INTEGRO-DIFFERENTIAL EQUATION OF FREDHOLM TYPE IN A COMPLEX PL

Abstract

1. Introduction

2. Preliminaries

3. Existence and Uniqueness

4. Numerical Study

5. Extended Study Overview

6. Simulation Results

Conclusion

References

Chapter 7LINEAR AND NONLINEAR PARTIALINTEGRO-DIFFERENTIAL EQUATIONSARISING FROM FINANCE

Abstract

1. Introduction

2. Background and Motivation

3. Preliminaries and Definitions

3.1. Exponential L´evy Models

3.2. Examples of L´evy Processes in Finance

3.2.1. Jump-Diffusion Models

Merton’s Model

3.2.2. Infinite Activity Pure Jump Models

Variance Gamma Process

Normal Inverse Gaussian Model

Generalized Hyperbolic Model

3.3. Admissible Activity L´evy Measures

4. Multidimensional Linear and Nonlinear PIDE

4.1. Existence and Uniqueness Results of PIDE

4.2. Maximal Monotone Operator Technique for SolvingNonlinear Parabolic Equations

4.2.1. Existence and Uniqueness of a Solution to the CauchyProblem

5. Applications to Option Pricing

5.1. Linearization of PIDE

6. Feedback Effects under Jump-Diffusion AssetPrice Dynamics

6.1. Numerical Simulation for the Underlying PIDE

7. Hamilton-Jacobi-Bellman Equation

7.1. Static Markowitz Model for Portfolio Optimization

7.2. Riccati Transformation of the HJB Equationand Application to Optimal Portfolio Selection Proble

7.2.1. Riccati Transformation

7.2.2. Properties of the Value Function as a Diffusion Function

7.2.3. Point-Wise a-Priori Estimates of Solution with Their Existenceand Uniqueness

7.2.4. Application to Stochastic Dynamic Optimal PortfolioSelection Problem

7.2.5. Numerical Examples

Conclusion

Acknowledgments

References

Index

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