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Abstract : Universiti Pendidikan Sultan Idris |
The research is aimed to find the most efficient implementation strategies by Gauss numerical
methods for solving stiff problems and the best error estimation in the variable
stepsize setting. The numerical methods considered as a research methodology are the 2-stage
(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods. Two strategies by Hairer and
Wanner (HW) and Gonzalez-Pinto, Montijano and Randez (GMR) schemes were implemented. The
variable stepsize setting employed the simplified Newton is modified to fit according to HW and
GMR schemes in solving the nonlinear algebraic systems of the equations. The error
estimation for the variable
stepsize setting is computed using extrapolation technique with stepsizes h and h 2 .
HW and GMR schemes used the transformation matrix T to improve the efficiency of the methods and
also compared with the modified Hairer and Wanner (MHW) scheme
without using any transformation matrix T . Findings showed that G2 method using
MHW scheme gave an efficient implementation in solving Kaps, Oreganator and HIRES
problems while for G3 method, it was efficient in solving Kaps, Brusselator, Oreganator, Van der
Pol and HIRES problems. In terms of error estimation, the G2 method gave the best error estimation
for Brusselator, Oreganator, Van der Pol and HIRES problems, while for the G3 method it was
efficient in solving Kaps, Brusselator, Oreganator, Van der Pol and HIRES problems, both by
using HW scheme. In conclusion, the MHW scheme without any transformation matrix T can be as
efficient as the HW and GMR schemes by using the variable stepsize setting and the MHW scheme is
recommended in solving stiff problems. As for the implications, this research could be extended
to other different types of problems such as delay and fuzzy
rential equations.
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