Introduction to integral equations with applications by Abdul Jarri

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Introduction to integral equations with applications written by Abdul Jarri, Clarkon University.

Introduction to integral equations with applications written by Abdul Jarri cover the following topics.

Integral Equations, Origin, and Basic Tools
Various Problems as Integral Equations, Exercises, Classification of Integral Equations, Exercises, Some Important Identities and Basic Definitions (Multiple Integrals Reduced to Single Integrals, Generalized Leibnitz Formula, Convergence of Integrals and Basic Definitions), Exercises, Laplace, Fourier, and Other Transforms (The Laplace Transform, Fourier Transforms, Other Transforms), Exercises, Basic Numerical Integration Formulas (Basic (Elementary) Integration Formulas, The Smoothing Effect of Integration, Interpolation ofthe Numerical Solutions of Integral Equations, Review of Cramer's Rule), Exercises

Modeling of Problems as Integral Equations
Population Dynamics (Human Population, Biological Species Living Together), Exercises, Control and Other Problems (Mortality of Equipment and Rate of Replacement), Exercises, Mechanics Problems (Hanging Chain, Sliding a Bead Along aWire: Abel's Problem), Exercises, Initial Value Problems Reduced to Volterra Integral Equations, Exercises, Boundary Value Problems Reduced to Fredholm Integral Equations, Exercises, Mixed Boundary Conditions: Dual Integral Equations (Electrified Infinite Plane, Electrified Disc), Exercises, Integral Equations in Higher Dimensions (Schrödinger Equation as an Integral Equation in the Three-Dimensional Momentum Space)

Volterra Integral Equations
Volterra Equations ofthe Second Kind, Resolvent Kernel Method: Neumann Series, Method of Successive Approximations (Iterations), Laplace Transform Method: Dijference Kernel, Volterra Integral Equations ofthe First Kind, Volterra Integral Equation ofthe First Kind with a Difference Kernel—Laplace Transform Method, Numerical Solution of Volterra Integral Equations, Numerical Approximation Setting of Volterra Equations, The Green''s Function

Construction ofthe Green's Function
Nonhomogeneous Differential Equations, Construction ofthe Green's Function — Variation of Parameters Method, Orthogonal Series Representation of Green 's Function, Green''s Function in Two Dimensions, Fredholm Integral Equations and the Green's Function, Fredholm Integral Equations 209

Fredholm Integral Equations with Degenerate Kernel
Nonhomogeneous Fredholm Equations with Degenerate Kernel, Fredholm Alternative, Approximating a Kernel by a Degenerate One, Fredholm Integral Equations with Symmetrie Kernel, Homogeneous Fredholm Equations with Symmetrie Kernel, Solution of Fredholm Equations ofthe Second Kind with Symmetrie Kernel, Fredholm Integral Equations ofthe Second Kind, Method of Fredholm Resolvent Kernel, Method of Iterated Kernels, Some Basic Approximate Methods, Fredholm Integral Equations of the First Kind, Fredholm Equations of the First Kind with Symmetrie Kernels, Ill-Posed Problems and the Fredholm Equation ofthe First Kind, Numerical Solution of Fredholm Integral Equations, Numerical Approximation Setting of Fredholm Integral Equations, Homogeneous Fredholm Equations

Existence ofthe Solutions: Basic Fixed Point Theorems
Preliminaries: Toward a Contractive Mapping, Basic Definitions: Complete Metrie Spaces, Contractive Mapping for Linear Fredholm Equations, Contractive Mapping for Linear Volterra, Fixed Point Theorem of Banach, Existence ofthe Solution for Linear Integral, Existence of the Solution for Nonlinear Integral Equations, Existence of the Solution for Nonlinear Differential Equations

Higher Quadrature Rules for the Numerical Solutions
Higher Quadrature Rules of Integration with Tables, Higher Quadrature Rules for Volterra Equations, Higher Quadrature Rules for Fredholm Equations, Comments on Higher Quadrature Rules for Some Singular Fredholm Equations

Appendix A The Hankel Transforms
A.l The Hankel Transform for the Electrified Disc, The Finite Hankel Transform

Appendix B Green 's Functionfor Various Boundary Value Problems
B.l Green 's Functions in Terms of Simple Functions, B.2 Green 's Function in Terms of Special Functions

Answers to Exercises

References

Index

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Worksheets (Solved)

Matrix Logarithm

Trigonometric Identities

Application of Trigonometry

Integration