Solving Fourth Order Differential Non-Linear Equations, Existence and Uniqueness

Abstract

The aim of this thesis is to present a novel numerical approach for the solution of a class of non-linear fourth-order boundary value problems. The method is based on embedding Green's function into some fixed point iteration schemes, an idea previously used in other works to investigate nonlinear boundary value problems of lower order. To this end, the thesis is divided in 6 chapters. The first chapter is a short description of the main ideas of this thesis. The second chapter represents a review on Green's functions for differential equations. In Chapter 3 some existence and uniqueness results for the fourth order boundary value problems are presented. The proposed numerical method is then explained and applied on numerical examples in Chapter 4, where a comparison with the Spline method is also given, demonstrating thus that our method yields accurate results up to 10 e-20, compared to the Spline method, where the results are accurate up to 10 e-13. The results obtained by our method were achieved within a reasonable time limit compared with other methods. Chapter 5 is concerned with the convergence analysis of our method. More precisely, some conditions which guarantees the convergence of the solution under specific conditions is given. For the proof we used the Banach-Picard theorem along with the Green's function. Finally, in Chapter 6 we present a short conclusions and a summary of the whole thesis.