A new class of orthogonal polynomials as trial function for the derivation of numerical integrators

This paper presents a set of newly constructed polynomials valid in interval [-1, 1] with respect to weight function w(x) = x + 1. For applicability sake, the polynomials shall be employed as trial function to develop a fast, efficient and reliable block algorithm for the numerical solution of ordinary differential equations with application to second order initial value problems. Collocation and interpolation techniques were adopted for the formulation of self-starting continuous hybrid schemes. Findings from the analysis of the basic properties of the method using appropriate existing theorems show that the developed schemes are consistent, zero-stable and hence convergent. On implementation, the superiority of the scheme over the existing method is established numerically. Further investigation of the properties of these polynomials is ongoing as we hope to discuss this in the future paper.