Collocation approximation for fourth-order boundary value problems

This paper concerns the numerical methods based on collocation approximation of fourth-order boundary value problems for the linear problem. Canonical polynomials constructed in [1] are modified and used as a new basis for a collocation solution via the perturbed collocation method with and without exponentially fitting. Nonlinear cases are treated by the Newton's linearization scheme of order 4. The Newton's scheme from the Taylor's series expansion of order 4 is given by: (1.0) G + Delta y partial derivative G/partial derivative y + Delta y' partial derivative G/partial derivative y' + Delta y '' partial derivative G/partial derivative y '' + Delta y''' partial derivative G/partial derivative y''' + Delta yiv partial derivative G/partial derivative yiv = 0, is used throughout this paper. Numerical examples are given to illustrate the effectiveness of the methods discussed in this paper.