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Type :thesis
Subject :QA Mathematics
Main Author :Sara Syahrunnisaa Mustapha
Title :An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
Place of Production :Tanjong Malim
Publisher :Fakulti Sains dan Matematik
Year of Publication :2021
Corporate Name :Universiti Pendidikan Sultan Idris
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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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