A Course on Integral Equations with Numerical Analysis. Advanced Numerical Analysis

✍ Scribed by Tofigh Allahviranloo, Armin Esfandiari

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 222
• Series: Mathematical Engineering
• Category: Library

✦ Table of Contents

Preface
Contents
1 Introduction to Numerical Analysis
1.1 Introduction
1.2 Error Analysis
1.2.1 Errors in an Algorithm
1.2.2 Round of Error and Floating Points Arithmetic
1.2.3 Algorithm Error Propagation
1.3 Interpolation
1.3.1 Lagrange Interpolation
1.3.2 Newton’s Divided Difference Interpolation
1.4 A Short Review on Vector Norms and Linear System of Equations
1.4.1 Vector Norm
1.4.2 Direct Methods
1.4.3 Numerical Methods
1.5 Numerical Integration
1.5.1 Newton–Cotes Integration Method
1.5.2 The Peano’s Kernel Representation
1.5.3 Gauss Integration Method
2 Interval Interpolation
2.1 Interval Error
2.1.1 Interval Calculations
2.2 Interval Interpolation
2.2.1 Theorem
2.2.2 Theorem
2.2.3 Theorem
2.2.4 Example
2.2.5 Example
3 Orthogonal Polynomials and Least Square Approximation
3.1 Orthogonal Polynomials
3.1.1 Definition-Inner Product of Definite Functions
3.1.2 Definition-Orthogonal Functions
3.1.3 Example
3.1.4 Example
3.1.5 Example
3.1.6 Definition
3.1.7 Example
3.1.8 Theorem
3.1.9 Orthogonal Polynomials and Least Squares Approximation
3.1.10 Example
3.1.11 Theorem
3.1.12 Example
4 Integral Equations
4.1 Introduction
4.1.1 Definition—Integral Equation
4.1.2 Example
4.1.3 Definition—First Type Integral Equation
4.1.4 Definition—Kernel
4.1.5 Definition—Homogeneous Integral Equation of the Second Type
4.1.6 Definition—Volterra Integral Equation
4.1.7 Definition—Integro-Differential Equation
4.1.8 Definition—The Integro-Differential Equation
4.1.9 The Relationship Between Integral Equations and Differential Equations
4.1.10 Lemma
4.1.11 Lemma
4.1.12 Lemma
4.1.13 Definition
4.2 Continuous Functions x(.) and L2
4.2.1 Definition
4.2.2 Definition
4.2.3 Definition
4.2.4 Definition
4.2.5 Cauchy–Schwarz Inequality Theorem
4.2.6 Theorem
4.2.7 Definition—Continuous Norm of a Continuous Kernel
4.2.8 Definition—Linear Operator
4.3 Production of Two Kernels
4.3.1 Lemma
4.3.2 Remark
4.3.3 Lemma
4.3.4 Definition
4.3.5 Fubini Theorem
4.3.6 Tonley–Hopson Theorem
4.3.7 Lemma
4.4 Fredholm Integral Equation of the Second Type
4.4.1 Definition—Regular Value
4.4.2 Theorem
4.4.3 Theorem
4.5 Continuous Kernels
4.5.1 Theorem
4.6 Adjoint Kernels
4.6.1 Definition
4.6.2 The Combination of Two Adjoint Functions
4.6.3 Definition—Normal Kernel
4.6.4 Remark
4.6.5 Adjoint Equations
4.6.6 Definition
4.6.7 Lemma
4.6.8 Remark
4.6.9 Theorem
4.6.10 Theorem
4.6.11 Example
4.6.12 Definition
4.6.13 Definition—Point Wise Convergence
4.6.14 Definition—Uniformly Convergence
4.6.15 Example
4.6.16 Definition
4.6.17 Definition
4.6.18 Definition
4.6.19 Theorem
4.6.20 Theorem
4.6.21 Theorem
4.6.22 Theorem
4.6.23 Theorem
4.6.24 Theorem
4.6.25 Theorem
5 Numerical Solution of Integral Equations
5.1 Introduction
5.2 Neumann Series
5.2.1 Theorem
5.2.2 Example
5.2.3 Error Calculation
5.3 Nystrom Method
5.3.1 Theorem
5.4 Gauss–Chebyshev Method
5.4.1 Chebyshev Expansion
5.4.2 Closed Gauss–Chebyshev Rule
5.4.3 Theorem
5.4.4 Disadvantages of the Gauss–Chebyshev Method
5.5 Non-singular Functions
5.5.1 Definition
5.5.2 Definition
5.5.3 Example
5.5.4 Example
5.5.5 Example
5.5.6 Example
5.6 Expansion Method
5.7 Collocation Methods
5.7.1 Example
5.8 Norm Chebyshev
5.9 Least Squares Method (L2-Norm Method)
5.9.1 Example
5.10 Numerical Solution of the Second Kind Integral Equations
6 Numerical Methods for Integral–Differential Equations
6.1 Introduction
6.2 Integral–Differential Equations
6.2.1 Example
6.3 El-Gendi Method
6.3.1 Example
6.4 Fast Galerkin Method
7 Introduction to Interval Integral Equations
7.1 Introduction
7.2 Interval Fredholm Integral Equations
7.2.1 Definition-Dual Interval System
7.2.2 Definition
7.2.3 Theorem
7.2.4 Remark
7.2.5 Definition: The Interval Number Vector
7.2.6 Definition
7.2.7 Definition
7.3 Interval Fredholm Integral Equation
7.3.1 Residual Minimization Method
References