HYBRID LINEAR MULTI-STEP METHODS FOR THE DIRECT SOLUTION OF INITIAL VALUE PROBLEMS OF THE SPECIAL THIRD ORDER ORDINARY DIFFERENTIAL EQUATIONS

The research encompasses the creation of hybrid linear multistep techniques geared toward solving special categories of third order ordinary differential equations (ODEs) directly. In order to help ease the burden created by previously provided solutions, like computation difficulties and absence of given starting conditions, the methods are formed through interpolation and collocation processes. Three block methods are formulated and defined, for one off grid points (TBHM1), two off grid points (TBHM2), and three off grid points (TBHM3). These methods are further explored and their order, consistency, zero stability, and convergence are scrutinized. The results from implementing these methods numerically outstrip the precision and rate of convergence of the other methods published in the rest of the literature. The methods, which are applied to particularly stiff third order linear problems, confirm the effectiveness of the methods since the results obtained are better than what is typical with ordinary approaches. This work enhances mathematical modeling by providing a self-starting solver for higher order ODEs which can be conveniently expanded to other systems. The methods have been developed in the study to prove higher order initial value problems in a direct approach system without having to tackle the transforms towards the first order systems thus showing that such problems can be proved. It can be said that the provided methods serve as powerful aids for the direct solution of the problems. The hybrid block methods derived in this study contribute greatly to mathematicians dealing with non orthogonal grids.