
EqWorld
The World of Mathematical Equations 

Information >
Mathematical Books >
Handbook of Integral Equations, Second Edition > Contents




Handbook of Integral Equations Second 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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) = g_{1}(x)h_{1}(t) + ... + g_{n}(x)h_{n}(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(x − t)
 10.5. Method of Fractional Differentiation
 10.6. Equations with Weakly Singular Kernel
 10.7. Method of Quadratures
 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) = g_{1}(x)h_{1}(t) + ... + g_{n}(x)h_{n}(t)
 11.3. Equations with Difference Kernel: K(x,t) = K(x − t)
 11.4. Operator Methods for Solving Linear Integral Equations
 11.5. Construction of Solutions of Integral Equations with Special RightHand 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
 11.10. Method of Quadratures
 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 IllPosed 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
 13.19. Quadrature 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 RingShaped 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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 PowerLaw 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
References
Index

The EqWorld website presents extensive information on solutions to
various classes of ordinary differential equations, partial differential
equations, integral equations, functional equations, and other mathematical
equations.
Copyright © 2008 Andrei D. Polyanin
