Computational Methods for Linear Integral Equations

Presents basic theoretical material on numerical analysis, convergence, error estimates and accuracy. The unique computational approach leads the reader from theoretical and practical problems to computation with hands-on guidance for input files and the execution of computer programs. All supporting Mathematicar files related to the book are available via the Internet at the authors' websites. For professionals, graduate students, and researchers in mathematics, physical sciences, and engineering. Readers interested in the numerical solution of integral equations will find the book's practical problem-solving style both accessible and useful for their work.

Includes supplementary material: sn.pub/extras

Klappentext

Integral equations have wide applications in various fields, including continuum mechanics, potential theory, geophysics, electricity and magnetism, kinetic theory of gases, hereditary phenomena in physics and biology, renewal theory, quantum mechanics, radiation, optimization, optimal control systems, communication theory, mathematical economics, population genetics, queueing theory, and medicine.

Computational Methods for Linear Integral Equations presents basic theoretical material that deals with numerical analysis, convergence, error estimates, and accuracy. The unique computational aspect leads the reader from theoretical and practical problems all the way through to computation with hands-on guidance for input files and the execution of computer programs.

Features:

• Offers all supporting Mathematica® files related to the book via the Internet at the authors' Web sites: www.math.uno.edu/fac/pkythe.html or www.math.uno.edu/fac/ppuri.html

• Contains identification codes for problems, related methods, and computer programs that are cross-referenced throughout the book to make the connections easy to understand

• Illustrates a how-to approach to computational work in the development of algorithms, construction of input files, timing, and accuracy analysis

• Covers linear integral equations of Fredholm and Volterra types of the first and second kinds as well as associated singular integral equations, integro-differential equations, and eigenvalue problems

• Provides clear, step-by-step guidelines for solving difficult and complex computational problems

  This book is an essential reference and authoritative resource for all professionals, graduate students, and researchers in mathematics, physical sciences, and engineering. Researchers interested in the numerical solution of integral equations will find its practical problem-solving style both accessible and useful for their work.

  Inhalt
  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.