Computing the positive solutions of the discrete third-order three-point right focal boundary-value problems

Abstract

The existence of positive solutions to the discrete third-order three-point boundary-value problems (BVPs) was recently established in Ji and Yang [Positive solutions of discrete third-order three-point right focal boundary value problems, J. Differ. Equat. Appl. 15 (2009), pp. 185–195]. In this paper, we propose an algorithm for the computation of such positive solutions. The method is based on the power method for the dominant eigenvalue and the Crout-like factorization algorithm for the sparse system of linear equations. At each iteration of the method, it calls for a linear solver with linear computational complexity. The proposed method is extremely effective for large-scale problems. A numerical example is also included to demonstrate the effectiveness of the algorithm when applied to the third-order three-point BVPs of differential equation.