A STUDY OF A BOUNDARY VALUE PROBLEM AND ITS EQUIVALENT FREDHOLM / VOLTERRA INTEGRAL EQUATION FORMULATION

Abstract

This dissertation presents a comprehensive study of linear second-order boundary value problems (BVPs) and their equivalent integral formulations. The model differential equa tion u′′(x) = x +1, 0 ≤ x ≤1, u(0)=0, u(1) = 0 is first solved analytically using classical ordinary differential equation techniques. The corresponding Green’s function for the differential operator is then derived, allowing the reformulation of the BVP as a Fredholm integral equation of the second kind. In addition, a singularly perturbed boundary value problem of the form εu′′(x) + a(x)u′(x) + b(x)u(x) = f(x), u(0) = u0, u(1) = u1 is considered, where ε is a small positive parameter. The study examines the behavior of the solution under perturbation and investigates whether the solution obtained from the differential equation is equivalent to its corresponding integral equation formulation. Both analytical and numerical methods are employed to evaluate the integral represen tations, and the resulting solutions are shown to be consistent with those obtained from the differential equations. All the numerical calculations in this paper were carried out in MATLAB using the solution of boundary value problems by numerical quadrature of the Fredholm integral equation, finite difference approximations and bvp4c,in solving ordinary differential equations. Comparison graphs between exact solution and numeri cal solutions from ODE and integral equation showed that their results agreed very well within the error of order of machine precision. We have showed that solutions of a boundary value problem posed in the form of differ ential equation and that posed in the form of integral equation agreed well when proper condition exists even in singular perturbation case, and that Green function can be a tool to transform boundary value problems into tractable integral equations.