Understanding Integro-Differential Equations

Name: Understanding Integro-Differential Equations
Brand: Nova Science Publishers Inc
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Vasundhara Devi J(Editor)

Nova Science Publishers Inc

Published on 18. October 2023

267 pages

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979-8-89113-136-1 (ISBN)

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Intro
UNDERSTANDING INTEGRODIFFERENTIALEQUATIONS
UNDERSTANDING INTEGRODIFFERENTIALEQUATIONS
Contents
PREFACE
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 PLANE
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 Problem
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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