Iterative Methods for the Numerical Solutions of Boundary Value Problems

Abstract

The aim of this thesis is twofold. First of all, in Chapters 1 and 2, we review the well-known Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM) for obtaining exact and numerical solutions for ordinary differential equations, partial differential equations, integral equations, integro-differential equations, delay differential equations, and algebraic equations in addition to calculus of variations problems. These schemes yield highly accurate solutions. However, local convergence is a main setback of such approaches. It means that the accuracy deteriorates as the specified domain becomes larger, that is as we move away from the initial conditions. Secondly, we present an alternative uniformly convergent iterative scheme that applies to an extended class of linear and nonlinear third order boundary value problems that arise in physical applications. The method is based on embedding Green's functions into well-established fixed point iterations, including Picard's and Krasnoselskii-Mann's schemes. The effectiveness of the proposed scheme is established by implementing it on several numerical examples, including linear and nonlinear third order boundary value problems. The resulting numerical solutions are compared with both the analytical and the numerical solutions that exist in the literature. From the results, it is observed that the present method approximates the exact solution very well and yields more accurate results than the ADM and the VIM. Finally, the numerical results confirm the applicability and superiority of the introduced method for tackling various nonlinear equations.