Hynichearry Justine (2018) Performance analysis of the family of conjugate gradient iterative methods with nonpolynomial spline scheme for solving second and fourthorder two point boundary value problems. Masters thesis, University Malaysia Sabah.

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Abstract
A numerical solution involving twopoint boundary value problems has vast contributions especially to formulate problems mathematically in fields such as science, engineering, and economics. In response to that, this study was conducted to solve for the secondand fourthorder twopoint boundary value problems (BVPs) by using cubic and quartic nonpolynomial spline discretization schemes for full, half and quartersweep cases. The derivation process based on the cubic and quartic nonpolynomial spline functions were implemented to generate the full, half and quartersweep cases nonpolynomial spline approximation equations. After that, the nonpolynomial spline approximation equations were used to generate the corresponding systems of linear equations in a matrix form. Since the systems of linear equations have large and sparse coefficient matrices, therefore the linear systems were solved by using the family of Conjugate Gradient (CG) iterative method. In order to conduct the performances comparative analysis of the CG iterative method, there are two other iterative methods were considered which are GaussSeide(l GS) and SuccessiveOverRelaxatio(nS OR)a long with the full, half and quartersweep concepts. Furthermore, the numerical experiments were demonstrated by solving three examples of second and fourthorder twopoint BVPs in order to investigate the performance analysis in terms of the number of iterations, execution time and maximum absolute error. Based on the numerical results obtained from the implementation of the three iteration families together with the cubic and quartic nonpolynomial spline schemes, the performance analysis of the CG iterative method was found to be superior to the GS and SOR iteration families in terms of the number of iteration, execution time and maximum absolute error when solving the twopoint BVPs. Hence, it can be stated that the CG iteration family is more efficient and accurate than the GS and SOR iteration families when solving the secondorder twopoint BVPs based on the cubic and quartic nonpolynomial spline schemes. However, for the fourthorder twopoint BVPs, the numerical results have shown that the implementation of the CG iteration family over the reduced system of secondorder twopoint BVPs failed to satisfy the convergence iteration criteria. As a result, the SOR iteration family is superior to GS iteration family in terms of the number of iteration, execution time and maximum absolute error.
Item Type:  Thesis (Masters) 

Keyword:  Boundary value problems (BVPs) , cubic and quartic nonpolynomial spline discretization schemes , Conjugate matrices 
Subjects:  Q Science > QA Mathematics 
Department:  FACULTY > Faculty of Science and Natural Resources 
Depositing User:  NORAINI LABUK  
Date Deposited:  12 Jul 2019 15:23 
Last Modified:  12 Jul 2019 15:23 
URI:  https://eprints.ums.edu.my/id/eprint/22608 
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