A Comparative Analysis of Some Numerical Methods for Integral Equations

Abstract

Integral and differential equations are the most tools used in building mathematical models for engineering and physical problems. The theory of integral equations has close contacts with many different areas of mathematics. Foremost among these are ordinary differential equations and systems of algebraic equations. In the area of differential equations many existence and uniqueness results has its origins in integral equations. Many Boundary Value Problems (BVP) can be formulated as integral equations of Fredholm type while Initial Value Problems (IVP) can be formulated as integral equations of Volterra type. The mathematical models represented in the form of integral equations have the advantages of considering local and global interactions, while those of differential equations involve only local interactions. Moreover, when solving the mathematical models numerically differential equation models give rise to large sparse systems while integral equation models give rise to dense coefficient matrices. The importance of the numerical solutions of integral and differential equations appears as a result of the difficulties with analytical solutions when used for real problems and also as a result of the great progress in computer tools. Also, it is interesting to remember that solutions of second order boundary value problems are elements in and solutions of fourth order boundary value problems are elements of while solutions of the corresponding Fredholm equations are elements in space. In this work, we consider the numerical solution of integral equations and in narrowing the scoop of the thesis we concentrate on integral equations which have their origins in differential equations especially those of the second and fourth order. Now if we have a physical problem which can be formulated as BVP and as integral equation, what is the best, from the numerical point of view, to solve the problem as a BVP or as integral equation provided that we use the same method and the same accuracy? Frist case: the problem is in the standard form of the second order twopoint boundary value problems and its integral representation. Second case: the problem is in the standard form of the fourth order twopoint boundary value problems and its integral representation. ii Third case: the problem contains some coefficients with physical parameters like those appears in the study of steady state convectiondiffusion problems. Fourth case the problem is in the general form of the second order twopoint boundary value problems with variable coefficients. It is found that the integral representations give more acceptable results than the differential equation representation, and the number of iteration in the integral form is less than that of the differential form. The thesis consists of three chapters: Chapter one: Basic concepts of integral and differential equations are considered and also, the relation between integral and differential equations with high concern on second and fourth order differential equations. Transformation of differential equation of higher order or system of differential equations of first order into one-dimensional integral equation using the replacement Lemma was studied. Relaxation iterative methods for solving linear algebraic equations and the choice of relaxation parameters to obtain the most rapid convergence are presented. As a measure and comparison of the speed of convergence of iterative methods we adopted an asymptotic convergence rate which depends on the spectral radius of the corresponding iteration matrix. Four approximate methods (the Liouville-Neumann series, the Laplace transform, the finite difference and the cubic Spline) for solving integral equations (Fredholm – Volterra) are discussed. Chapter two: This chapter is dedicated to BVP of the second order in the form: ( ) ( ) ( ) ( ) ( ) ( ) and fourth order BVP of the form ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) With the help of the replacement lemma, a single integral equation of Fredholm type, ( ) ( ) ∫ ( ) ( ) , corresponding to the given boundary value problem is obtained. The case of writing the fourth order boundary value as a system of two second order boundary value problems is considered. Also, the general linear second order BVP ( ) ( ) ( ) ( ) ( ) ( ) , with its