This paper is concerned with the approximate solution of third order boundary value problems by means of fourth order polynomial B-splines. For a given problem in a compact interval the authors consider a uniform grid \(\pi\) on it and look for approximate solutions of the form \( v(x) = \sum_{j=-2}^n C_j B_j(x)\) where the \(B_j\) are the elements of the fourth degree B-spline basis on \( \pi\). Now by imposing the collocation conditions of the approximate solution on some points they arrive to a set of algebraic equations which determine the \(C_j\), provided that this set possess a unique solution its computation provides the approximate B-spline solution.
Two numerical examples with linear and nonlinear equations are presented to show the applicability of the proposed approach. Some instabilities generated by the collocation conditions are detected and a partial remedy to avoid this problem is proposed.