A class of hybrid collocation methods for third-order ordinary differential equations

Abstract

A class of numerical methods is proposed for solving general third-order ordinary differential equations directly by collocation at the grid points x = x n+j , i = 0(1)k and at an off grid point x = x n+u , where k is the step number of the method and u is an arbitrary rational number in (x n , x n+k ). A predictor of order 2k − 1 is also proposed to cater for y n+k in the main method. Taylor series expansion is employed for the calculation of y n+1, y n+2, y n+u and their higher derivatives. Evaluation of the resulting method at x = x n+k for any value of u in the specified open interval yields a particular discrete scheme as a special case of the method. The efficiency of the method is tested on some general initial value problems of third-order ordinary differential equations.