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Full Description
This book presents numerical methods and computational aspects for linear integral equations. Such equations occur in various areas of applied mathematics, physics, and engineering. The material covered in this book, though not exhaustive, offers useful techniques for solving a variety of problems. Historical information cover ing the nineteenth and twentieth centuries is available in fragments in Kantorovich and Krylov (1958), Anselone (1964), Mikhlin (1967), Lonseth (1977), Atkinson (1976), Baker (1978), Kondo (1991), and Brunner (1997). Integral equations are encountered in a variety of applications in many fields including continuum mechanics, potential theory, geophysics, electricity and mag netism, kinetic theory of gases, hereditary phenomena in physics and biology, renewal theory, quantum mechanics, radiation, optimization, optimal control sys tems, communication theory, mathematical economics, population genetics, queue ing theory, and medicine. Most of the boundary value problems involving differ ential equations can be converted into problems in integral equations, but there are certain problems which can be formulated only in terms of integral equations. A computational approach to the solution of integral equations is, therefore, an essential branch of scientific inquiry.
Contents
1 Introduction.- 1 1 Notation and Definitions.- 1.2 Classification.- 1.3 Function Spaces.- 1.4 Convergence.- 1.5 Inverse Operator.- 1.6 Nyström System.- 1.7 Other Types of Kernels.- 1.8 Neumann Series.- 1.9 Resolvent Operator.- 1.10 Fredholm Alternative.- 2 Eigenvalue Problems.- 2.1 Linear Symmetric Equations.- 2.2 Residual Methods.- 2.3 Degenerate Kernels.- 2.4 Replacement by a Degenerate Kernel.- 2.5 Baterman's Method.- 2.6 Generallized Eigenvalue Problem.- 2.7 Applications.- 3 Equations of the Second Kind.- 3.1 Fredholm Equations.- 3.2 Volterra Equations.- 4 Classical Methods for FK2.- 4.1 Expansion Method.- 4.2 Product-Integration Method.- 4.3 Quadrature Method.- 4.4 Deferred Correction Methods.- 4.5 A Modified Quadrature Method.- 4.6 Collocation Methods.- 4.7 Elliott's Modification.- 5 Variational Methods.- 5.1 Galerkin Method.- 5.2 Ritz-Galerkin Methods.- 5.3 Special Cases.- 5.4 Fredholm-Nyström System.- 6 Iteration Methods.- 6.1 Simple Iterations.- 6.2 Quadrature Formulas.- 6.3 Error Analysis.- 6.4 Iterative Scheme.- 6.5 Krylov-Bogoliubov Method.- 7 Singular Equations.- 7.1 Singularities in Linear Equations.- 7.2 Fredholm Theorems.- 7.3 Modified Quadrature Rule.- 7.4 Convolution-Type Kernels.- 7.5 Volterra-Type Singular Equations.- 7.6 Convolution Methods.- 7.7 Asymptotic Methods for Log-Singular Equations.- 7.8 Iteration Methods.- 7.9 Singular Equations with the Hilbert Kernel.- 7.10 Finite-Part Singular Equations.- 8 Weakly Singular Equations.- 8.1 Weakly Singular Kernel.- 8.2 Taylor's Series Method.- 8.3 Lp-Approximation Method.- 8.4 Product-Integration Method.- 8.5 Splines Method.- 8.6 Weakly Singular Volterra Equations.- 9 Cauchy Singular Equations.- 9.1 Cauchy Singular Equations of the First Kind.- 9.2 Approximation by Trigonometric Polynomials.-9.3 Cauchy Singular Equations of the Second Kind.- 9.4 From CSK2 to FK2.- 9.5 Gauss-Jacobi Quadrature.- 9.6 Collocation Method for CSK1.- 10 Sinc-Galerkin Methods.- 10.1 Sine Function Approximations.- 10.2 Conformal Maps and Interpolation.- 10.3 Approximation Theory.- 10.4 Convergence.- 10.5 Sinc-Galerkin Scheme.- 10.6 Computation Guidelines.- 10.7 Sine-Collocation Method.- 10.8 Single-Layer Potential.- 10.9 Double-Layer Problem.- 11 Equations of the First Kind.- 11.1 Inherent Ill-Posedness.- 11.2 Separable Kernels.- 11.3 Some Theorems.- 11.4 Numerical Methods.- 11.5 Volterra Equations of the First Kind.- 11.6 Abel's Equation.- 11.7 Iterative Schemes.- 12 Inversion of Laplace Transforms.- 12.1 Laplace Transforms.- 12.2 General Interpolating Scheme.- 12.3 Inversion by Fourier Series.- 12.4 Inversion by the Riemann Sum.- 12.5 Approximate Formulas.- A Quadrature Rules.- A. 1 Newton-Cotes Quadratures.- A.2 Gaussian Quadratures.- A.3 Integration of Products.- A.4 Singular Integrals.- A.5 Infinite-Range Integrals.- A. 6 Linear Transformation of Quadratures.- A.7 Trigonometric Polynomials.- A.8 Condition Number.- A.7 Quadrature Tables.- B Orthogonal Polynomials.- B.l Zeros of Some Orthogonal Polynomials.- C Whittaker's Cardinal Function.- C. 1 Basic Results.- C.2 Approximation of an Integral.- D Singular Integrals.- D.l Cauchy's Principal-Value Integrals.- D.2 P.V. of a Singular Integral on a Contour.- D.3 Hadamard's Finite-Part Integrals.- D.4 Two-Sided Finite-Part Integrals.- D.5 One-Sided Finite-Part Integrals.- D.6 Examples of Cauchy P.V. Integrals.- D.7 Examples of Hadamard's Finite-Part Integrals.
