 EqWorld The World of Mathematical Equations  ## Handbook of Integral EquationsSecond Edition, Updated, Revised and Extended

Publisher: Chapman & Hall/CRC Press
Publication Date: 14 February 2008
Number of Pages: 1144

## Contents

• Authors
• Preface
• Some Remarks and Notation

### Part I. Exact Solutions of Integral Equations

#### 1. Linear Equations of the First Kind with Variable Limit of Integration

• 1.1. Equations Whose Kernels Contain Power-Law Functions
• 1.2. Equations Whose Kernels Contain Exponential Functions
• 1.3. Equations Whose Kernels Contain Hyperbolic Functions
• 1.4. Equations Whose Kernels Contain Logarithmic Functions
• 1.5. Equations Whose Kernels Contain Trigonometric Functions
• 1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions
• 1.7. Equations Whose Kernels Contain Combinations of Elementary Functions
• 1.8. Equations Whose Kernels Contain Special Functions
• 1.9. Equations Whose Kernels Contain Arbitrary Functions
• 1.10. Some Formulas and Transformations

#### 2. Linear Equations of the Second Kind with Variable Limit of Integration

• 2.1. Equations Whose Kernels Contain Power-Law Functions
• 2.2. Equations Whose Kernels Contain Exponential Functions
• 2.3. Equations Whose Kernels Contain Hyperbolic Functions
• 2.4. Equations Whose Kernels Contain Logarithmic Functions
• 2.5. Equations Whose Kernels Contain Trigonometric Functions
• 2.6. Equations Whose Kernels Contain Inverse Trigonometric Functions
• 2.7. Equations Whose Kernels Contain Combinations of Elementary Functions
• 2.8. Equations Whose Kernels Contain Special Functions
• 2.9. Equations Whose Kernels Contain Arbitrary Functions
• 2.10. Some Formulas and Transformations

#### 3. Linear Equations of the First Kind with Constant Limits of Integration

• 3.1. Equations Whose Kernels Contain Power-Law Functions
• 3.2. Equations Whose Kernels Contain Exponential Functions
• 3.3. Equations Whose Kernels Contain Hyperbolic Functions
• 3.4. Equations Whose Kernels Contain Logarithmic Functions
• 3.5. Equations Whose Kernels Contain Trigonometric Functions
• 3.6. Equations Whose Kernels Contain Combinations of Elementary Functions
• 3.7. Equations Whose Kernels Contain Special Functions
• 3.8. Equations Whose Kernels Contain Arbitrary Functions
• 3.9. Dual Integral Equations of the First Kind

#### 4. Linear Equations of the Second Kind with Constant Limits of Integration

• 4.1. Equations Whose Kernels Contain Power-Law Functions
• 4.2. Equations Whose Kernels Contain Exponential Functions
• 4.3. Equations Whose Kernels Contain Hyperbolic Functions
• 4.4. Equations Whose Kernels Contain Logarithmic Functions
• 4.5. Equations Whose Kernels Contain Trigonometric Functions
• 4.6. Equations Whose Kernels Contain Inverse Trigonometric Functions
• 4.7. Equations Whose Kernels Contain Combinations of Elementary Functions
• 4.8. Equations Whose Kernels Contain Special Functions
• 4.9. Equations Whose Kernels Contain Arbitrary Functions
• 4.10. Some Formulas and Transformations

#### 5. Nonlinear Equations of the First Kind with Variable Limit of Integration

• 5.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
• 5.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
• 5.3. Equations with Nonlinearity of General Form

#### 6. Nonlinear Equations of the Second Kind with Variable Limit of Integration

• 6.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
• 6.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
• 6.3. Equations with Power-Law Nonlinearity
• 6.4. Equations with Exponential Nonlinearity
• 6.5. Equations with Hyperbolic Nonlinearity
• 6.6. Equations with Logarithmic Nonlinearity
• 6.7. Equations with Trigonometric Nonlinearity
• 6.8. Equations with Nonlinearity of General Form

#### 7. Nonlinear Equations of the First Kind with Constant Limits of Integration

• 7.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
• 7.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
• 7.3. Equations with Power-Law Nonlinearity That Contain Arbitrary Functions
• 7.4. Equations with Nonlinearity of General Form

#### 8. Nonlinear Equations of the Second Kind with Constant Limits of Integration

• 8.1. Equations with Quadratic Nonlinearity That Contain Arbitrary Parameters
• 8.2. Equations with Quadratic Nonlinearity That Contain Arbitrary Functions
• 8.3. Equations with Power-Law Nonlinearity
• 8.4. Equations with Exponential Nonlinearity
• 8.5. Equations with Hyperbolic Nonlinearity
• 8.6. Equations with Logarithmic Nonlinearity
• 8.7. Equations with Trigonometric Nonlinearity
• 8.8. Equations with Nonlinearity of General Form

### Part II. Methods for Solving Integral Equations

#### 9. Main Definitions and Formulas. Integral Transforms

• 9.1. Some Definitions, Remarks, and Formulas
• 9.2. Laplace Transform
• 9.3. Mellin Transform
• 9.4. Fourier Transform
• 9.5. Fourier Cosine and Sine Transforms
• 9.6. Other Integral Transforms

#### 10. Methods for Solving Linear Equations of the Form K(x,t) y(t) dt = f(x)

• 10.1. Volterra Equations of the First Kind
• 10.2. Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + ... + gn(x)hn(t)
• 10.3. Reduction of Volterra Equations of the First Kind to Volterra Equations of the Second Kind
• 10.4. Equations with Difference Kernel: K(x,t) = K(xt)
• 10.5. Method of Fractional Differentiation
• 10.6. Equations with Weakly Singular Kernel
• 10.8. Equations with Infinite Integration Limit

#### 11. Methods for Solving Linear Equations of the Form y(x) − K(x,t) y(t) dt = f(x)

• 11.1. Volterra Integral Equations of the Second Kind
• 11.2. Equations with Degenerate Kernel: K(x,t) = g1(x)h1(t) + ... + gn(x)hn(t)
• 11.3. Equations with Difference Kernel: K(x,t) = K(xt)
• 11.4. Operator Methods for Solving Linear Integral Equations
• 11.5. Construction of Solutions of Integral Equations with Special Right-Hand Side
• 11.6. Method of Model Solutions
• 11.7. Method of Differentiation for Integral Equations
• 11.8. Reduction of Volterra Equations of the Second Kind to Volterra Equations of the First Kind
• 11.9. Successive Approximation Method
• 11.11. Equations with Infinite Integration Limit

#### 12. Methods for Solving Linear Equations of the Form K(x,t) y(t) dt = f(x)

• 12.1. Some Definition and Remarks
• 12.2. Integral Equations of the First Kind with Symmetric Kernel
• 12.3. Integral Equations of the First Kind with Nonsymmetric Kernel
• 12.4. Method of Differentiation for Integral Equations
• 12.5. Method of Integral Transforms
• 12.6. Krein's Method and Some Other Exact Methods for Integral Equations of Special Types
• 12.7. Riemann Problem for the Real Axis
• 12.8. Carleman Method for Equations of the Convolution Type of the First Kind
• 12.9. Dual Integral Equations of the First Kind
• 12.10. Asymptotic Methods for Solving Equations with Logarithmic Singularity
• 12.11. Regularization Methods
• 12.12. Fredholm Integral Equation of the First Kind as an Ill-Posed Problem

#### 13. Methods for Solving Linear Equations of the Form y(x) − K(x,t) y(t) dt = f(x)

• 13.1. Some Definition and Remarks
• 13.2. Fredholm Equations of the Second Kind with Degenerate Kernel. Some Generalizations
• 13.3. Solution as a Power Series in the Parameter. Method of Successive Approximations
• 13.4. Method of Fredholm Determinants
• 13.5. Fredholm Theorems and the Fredholm Alternative
• 13.6. Fredholm Integral Equations of the Second Kind with Symmetric Kernel
• 13.7. Integral Equations with Nonnegative Kernels
• 13.8. Operator Method for Solving Integral Equations of the Second Kind
• 13.9. Methods of Integral Transforms and Model Solutions
• 13.10. Carleman Method for Integral Equations of Convolution Type of the Second Kind
• 13.11. Wiener–Hopf Method
• 13.12. Krein's Method for Wiener–Hopf Equations
• 13.13. Methods for Solving Equations with Difference Kernels on a Finite Interval
• 13.14. Method of Approximating a Kernel by a Degenerate One
• 13.15. Bateman Method
• 13.16. Collocation Method
• 13.17. Method of Least Squares
• 13.18. Bubnov–Galerkin Method
• 13.20. Systems of Fredholm Integral Equations of the Second Kind
• 13.21. Regularization Method for Equations with Infinite Limits of Integration

#### 14. Methods for Solving Singular Integral Equations of the First Kind

• 14.1. Some Definitions and Remarks
• 14.2. Cauchy Type Integral
• 14.3. Riemann Boundary Value Problem
• 14.4. Singular Integral Equations of the First Kind
• 14.5. Multhopp–Kalandiya Method
• 14.6. Hypersingular Integral Equations

#### 15. Methods for Solving Complete Singular Integral Equations

• 15.1. Some Definitions and Remarks
• 15.2. Carleman Method for Characteristic Equations
• 15.3. Complete Singular Integral Equations Solvable in a Closed Form
• 15.4. Regularization Method for Complete Singular Integral Equations
• 15.5. Analysis of Solutions Singularities for Complete Integral Equations with Generalized Cauchy Kernels
• 15.6. Direct Numerical Solution of Singular Integral Equations with Generalized Kernels

#### 16. Methods for Solving Nonlinear Integral Equations

• 16.1. Some Definitions and Remarks
• 16.2. Exact Methods for Nonlinear Equations with Variable Limit of Integration
• 16.3. Approximate and Numerical Methods for Nonlinear Equations with Variable Limit of Integration
• 16.4. Exact Methods for Nonlinear Equations with Constant Integration Limits
• 16.5. Approximate and Numerical Methods for Nonlinear Equations with Constant Integration Limits
• 16.6. Existence and Uniqueness Theorems for Nonlinear Equations
• 16.7. Nonlinear Equations with a Parameter: Eigenfunctions, Eigenvalues, Bifurcation Points

#### 17. Methods for Solving Multidimensional Mixed Integral Equations

• 17.1. Some Definition and Remarks
• 17.2. Methods of Solution of Mixed Integral Equations on a Finite Interval
• 17.3. Methods of Solving Mixed Integral Equations on a Ring-Shaped Domain
• 17.4. Projection Method for Solving Mixed Equations on a Bounded Set

#### 18. Application of Integral Equations for the Investigation of Differential Equations

• 18.1. Reduction of the Cauchy Problem for ODEs to Integral Equations
• 18.2. Reduction of Boundary Value Problems for ODEs to Volterra Integral Equations. Calculation of Eigenvalues
• 18.3. Reduction of Boundary Value Problems for ODEs to Fredholm Integral Equations with the Help of the Green's Function
• 18.4. Reduction of PDEs with Boundary Conditions of the Third Kind to Integral Equations
• 18.5. Representation of Linear Boundary Value Problems in Terms of Potentials
• 18.6. Representation of Solutions of Nonlinear PDEs in Terms of Solutions of Linear Integral Equations (Inverse Scattering)

### Supplements

#### Supplement 1. Elementary Functions and Their Properties

• 1.1. Power, Exponential, and Logarithmic Functions
• 1.2. Trigonometric Functions
• 1.3. Inverse Trigonometric Functions
• 1.4. Hyperbolic Functions
• 1.5. Inverse Hyperbolic Functions

#### Supplement 2. Finite Sums and Infinite Series

• 2.1. Finite Numerical Sums
• 2.2. Finite Functional Sums
• 2.3. Infinite Numerical Series
• 2.4. Infinite Functional Series

#### Supplement 3. Tables of Indefinite Integrals

• 3.1. Integrals Involving Rational Functions
• 3.2. Integrals Involving Irrational Functions
• 3.3. Integrals Involving Exponential Functions
• 3.4. Integrals Involving Hyperbolic Functions
• 3.5. Integrals Involving Logarithmic Functions
• 3.6. Integrals Involving Trigonometric Functions
• 3.7. Integrals Involving Inverse Trigonometric Functions

#### Supplement 4. Tables of Definite Integrals

• 4.1. Integrals Involving Power-Law Functions
• 4.2. Integrals Involving Exponential Functions
• 4.3. Integrals Involving Hyperbolic Functions
• 4.4. Integrals Involving Logarithmic Functions
• 4.5. Integrals Involving Trigonometric Functions
• 4.6. Integrals Involving Bessel Functions

#### Supplement 5. Tables of Laplace Transforms

• 5.1. General Formulas
• 5.2. Expressions with Power-Law Functions
• 5.3. Expressions with Exponential Functions
• 5.4. Expressions with Hyperbolic Functions
• 5.5. Expressions with Logarithmic Functions
• 5.6. Expressions with Trigonometric Functions
• 5.7. Expressions with Special Functions

#### Supplement 6. Tables of Inverse Laplace Transforms

• 6.1. General Formulas
• 6.2. Expressions with Rational Functions
• 6.3. Expressions with Square Roots
• 6.4. Expressions with Arbitrary Powers
• 6.5. Expressions with Exponential Functions
• 6.6. Expressions with Hyperbolic Functions
• 6.7. Expressions with Logarithmic Functions
• 6.8. Expressions with Trigonometric Functions
• 6.9. Expressions with Special Functions

#### Supplement 7. Tables of Fourier Cosine Transforms

• 7.1. General Formulas
• 7.2. Expressions with Power-Law Functions
• 7.3. Expressions with Exponential Functions
• 7.4. Expressions with Hyperbolic Functions
• 7.5. Expressions with Logarithmic Functions
• 7.6. Expressions with Trigonometric Functions
• 7.7. Expressions with Special Functions

#### Supplement 8. Tables of Fourier Sine Transforms

• 8.1. General Formulas
• 8.2. Expressions with Power-Law Functions
• 8.3. Expressions with Exponential Functions
• 8.4. Expressions with Hyperbolic Functions
• 8.5. Expressions with Logarithmic Functions
• 8.6. Expressions with Trigonometric Functions
• 8.7. Expressions with Special Functions

#### Supplement 9. Tables of Mellin Transforms

• 9.1. General Formulas
• 9.2. Expressions with Power-Law Functions
• 9.3. Expressions with Exponential Functions
• 9.4. Expressions with Logarithmic Functions
• 9.5. Expressions with Trigonometric Functions
• 9.6. Expressions with Special Functions

#### Supplement 10. Tables of Inverse Mellin Transforms

• 10.1. Expressions with Power-Law Functions
• 10.2. Expressions with Exponential and Logarithmic Functions
• 10.3. Expressions with Trigonometric Functions
• 10.4. Expressions with Special Functions

#### Supplement 11. Special Functions and Their Properties

• 11.1. Some Coefficients, Symbols, and Numbers
• 11.2. Error Functions. Exponential and Logarithmic Integrals
• 11.3. Sine Integral and Cosine Integral. Fresnel Integrals
• 11.4. Gamma Function, Psi Function, and Beta Function
• 11.5. Incomplete Gamma and Beta Functions
• 11.6. Bessel Functions (Cylindrical Functions)
• 11.7. Modified Bessel Functions
• 11.8. Airy Functions
• 11.9. Confluent Hypergeometric Functions
• 11.10. Gauss Hypergeometric Functions
• 11.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions
• 11.12. Parabolic Cylinder Functions
• 11.13. Elliptic Integrals
• 11.14. Elliptic Functions
• 11.15. Jacobi Theta Functions
• 11.16. Mathieu Functions and Modified Mathieu Functions
• 11.17. Orthogonal Polynomials
• 11.18. Nonorthogonal Polynomials

#### Supplement 12. Some Notions of Functional Analysis

• 12.1. Functions of Bounded Variation
• 12.2. Stieltjes Integral
• 12.3. Lebesgue Integral
• 12.4. Linear Normed Spaces
• 12.5. Euclidean and Hilbert Spaces. Linear Operators in Hilbert Spaces

#### Index

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