A P-Stable Linear Multistep Method For Solving General Third Order Ordinary Differential Equations

A P-stable linear multistep method for solving general third order initial value problems of ordinary differential equations without first reducing the problems into a system of first order equations is considered. The approach for the development of this method is essentially based on collocation of the differential system generated from a basis function. A predictor for the evaluation of $y_{n+k}$ for an odd $k\ge 3$ in the main method is also proposed. The two resulting methods, the corrector and the predictor are P-stable for $k = 3$ . These as a block are tested on a number of problems to show their efficiency. When the methods (the corrector and the predictor) are evaluated at $x = x_{n+3}$ identical schemes are obtained as special cases of the methods, while the set of first and second derivatives obtained from the corrector are different from those obtained from the predictor.