One-Step Hybrid Block Method Containing Third Derivatives and Improving Strategies for Solving Bratu's and Troesch's Problems

Abstract

In this paper, we develop a one-step hybrid block method for solving boundary value problems, which is applied to the classical one-dimensional Bratu's and Troesch's problems. The convergence analysis of the new technique is discussed, and some improving strategies are considered to get better performance of the method. The proposed approach produces discrete approximations at the grid points, obtained after solving an algebraic system of equations. The solution of this system is obtained through a homotopy-type strategy used to provide the starting points needed by Newton's method. Some numerical experiments are presented to show the performance and effectiveness of the proposed approach in comparison with other methods that appeared in the literature.