An Improved Order Seven Hybrid-Method for the Integration of Stiff First-Order Differential Equations

This research deals with the derivation of an improved two-third step hybrid-block approach for the numerical integration of stiff first-order ordinary differential equations with initial values. The method derived from the collocation and interpolation of the basis function (power series) gives rise to a continuous implicit method. The estimation approach at the off-grid points gives rise to the continuous implicit linear multistep method. The evaluation of the continuous method at various points yields the block method. We investigated the following basic features: the order and error constant, zero stability, consistency, and convergence of the block method to test its efficiency. The new block method was used to solve some stiff problems to generate more efficient results when matched with some existing authors solving the same stiff first-order problems. Thus, the method can be used as an effective tool to solve first-order stiff problems.