Hybrid Predictors Methods containing Third Derivatives for Solving Stiff and Non-Stiff First-order Differential Equations

We present a series of hybrid predictor-based third derivative linear multistep algorithms for solving stiff initial value problems in ordinary differential equations. The third-order derivative methods were developed using power series collocation and interpolation techniques. New approaches' stability properties were examined and presented. The new schemes' region of absolute stability, as measured by the boundary locus approach, is found to be A-stable. The methods were tested on some stiff problems and numerical results and errors obtained by using the new schemes were found to compare favourably with some existing methods.