Performance analysis of the family of conjugate gradient iterative methods with non-polynomial spline scheme for solving second- and fourth-order two point boundary value problems - Hynichearry Justine (2018)
A numerical solution involving two-point 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 fourth-order two-point boundary value problems (BVPs) by using cubic and quartic non-polynomial spline discretization schemes for full-, half- and quarter-sweep cases. The derivation process based on the cubic and quartic non-polynomial spline functions were implemented to generate the full-, half- and quarter-sweep cases non-polynomial spline approximation equations. After that, the non-polynomial 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 Gauss-Seide(l GS) and Successive-Over-Relaxatio(nS OR)a long with the full-, half- and quarter-sweep concepts. Furthermore, the numerical experiments were demonstrated by solving three examples of second- and fourth-order two-point 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 non-polynomial 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 two-point 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 two-point BVPs based on the cubic and quartic non-polynomial spline schemes. However, for the fourth-order two-point BVPs, the numerical results have shown that the implementation of the CG iteration family over the reduced system of second-order two-point 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.