Numerical Solutions of Fredholm Integral Equations of the Second Kind Research Report in Mathematics, Number 30, 2020

Abstract

This thesis mainly focuses on the Mathematical and Numerical aspects of the Fredholm integral equations of the Second Kind. Due to its wide range of physical applications, we are going to deal with three types of equations namely: differential equations, integral equations and integro-differential equations. Some of the applications of integral equations are heat conducting radiation, elasticity, potential theory and electrostatics. Generally, we define an integral equation where an unknown function occurs under an integral sign. Also integral equations can be classified according to three different dichotomies: 1. Nature of the limits of integration 2. Placement of the unknown function 3. Nature of the known function After the classification of these integral equations we will have to investigate some analytical and numerical methods for solving the Fredholm integral equations of the Second Kind. Analytical methods include: degenerate kernel methods, Adomain decomposition methods and Successive approximation methods and Numerical Methods include: Degenerate kernel methods, Projection methods, Nystrom methods and Spectral methods. The main objective of the thesis is to study Fredholm integral equations of the Second Kind. In chapter 4 we have given the approximate methods (Spectral Methods) to solve these equations Using The Classical Orthogonal polynomials which is the main idea of this thesis where we apply the Spectral Approximation Methods for approximating the Fredholm integral equations of the Second Kind.