
Understanding Integro-Differential Equations
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Content
- 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
- Blank Page
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