DERIVATION OF BLOCK METHODS USING CONTINUOUS FORMULATION OF THE EXTENDED TRAPEZOIDAL RULE OF SECOND KIND FOR THE SOLUTION OF STIFF ORDINARY DIFFERENTIAL EQUATIONS

Yohanna Sani Awari (awari04c@yahoo.com)
Department of Mathematics, University of Jos
March, 2017
Full text (external site)
 

Abstract

The study focused on the derivation and implementation of Block Extended Trapezoidal Rule of Second Kind method and its hybrid form for step numbers k=3,5,…,13 for the solution of stiff ordinary differential equations. In this study, the multistep collocation approach was employed, which yields continuous formulation to derive Block Extended Trapezoidal Rule of Second Kind (BETR2s) and the Block Hybrid Extended Trapezoidal Rule of Second Kind methods (BHETR2s) to overcome the problems associated with the standard Extended Trapezoidal Rule of Second Kind methods (ETR2s). The discrete schemes used in the block form were efficiently obtained from their respective continuous formulation. This block approach is self-starting for k≥2 during the implementation stage, this circumvents both the non-self starting property associated with the standard ETR2s. Convergence analysis of the new block methods were carried out and results showed that all the new block methods are consistent and zero-stable, hence convergent. All the newly derived block methods were shown to be of order p=k+1 for the BETR2s case and order p=k+2 for the BHETR2s case. The absolute stability regions of the new methods plotted indicate that they possess regions suitable for the solution of stiff ordinary differential equations. The newly derived block methods were implemented on eight stiff systems of ordinary differential equations occurring in real life. With increase in the order, the block hybrid ETR2s (BHETR2s) methods have shown better accuracy than both our block ETR2s and the ETR2s using the boundary value method. We also observed that both BETR2s and BHETR2s compete favourably with the Ode Solvers. This research work is recommended to programmers to develop computer software(s). It is also recommended for the solution of partial differential equations via the Method of Lines (MOL).