A Collocation Technique Based on Orthogonal Polynomials for Construction of Continuous Hybrid Methods

Abstract

The paper focuses on formulation of a number of algorithms for the numerical solution of first order ordinary differential equations with applications to initial value problems. For this purpose, an orthogonal polynomial valid in interval [-1,1] and with respect to weight function w(x)=x­­2 was constructed and employed as basis function for the development of some continuous hybrid schemes in a collocation and interpolation technique. To make the continuous schemes self-starting, some block methods of discrete hybrid form were derived. The schemes were analyzed using appropriate existing theorems to investigate their stability, consistency and convergence. The investigation shows that the developed schemes are consistent, zero-stable and hence convergent. The self-starting methods were implemented on some test problems from the literature to show the accuracy and effectiveness of the schemes.