A Two-step Hybrid Block Method with Four Off-step Points on Singular Initial and Boundary Value Problems

Abstract

This work contains the numerical studies of singular initial and boundary value problems of ordinary differential equations. The importance of these problems arose from the recent attentions Mathematicians have drawn to them due to their occurrence naturally and repeatedly in physical models. The numerical solutions to these problems are obtained by a class of block hybrid methods. The behaviour of basis functions differs and as such has an influence on numerically derived methods. Previous studies eulogize the use of a single basis function. This study establishes the advantage of using multiple basis functions as the derived methods combine multiple properties of basis function as this approach improves the stability of the resulting methods. This work combines two basis functions namely: a newly Constructed orthogonal polynomial and shifted Chebyshev orthogonal polynomial for the development of some continuous hybrid schemes in 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 definitions to investigate their stability, consistency, and convergence. The investigation shows that the developed schemes are consistent, zero stable, and hence convergent. Comparison with the exact solutions and existing methods show that the proposed methods are effective numerical methods for the solutions of singular initial and boundary value problems.

Keywords

• Singular boundary value problems
• Orthogonal polynomial
• Stability
• consistence and convergence
• hybrid point
• basis function

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