Abstract. Solving two-point boundary value problems has become a scope of interest among many researchers due to its significant contributions in the field of science, engineering, and economics which is evidently apparent in many previous literary publications. This present paper aims to discretize the two-point boundary value problems by using a quartic non-polynomial spline before finally solving them iteratively with Conjugate Gradient (CG) method. Then, the performances of the proposed approach in terms of iteration number, execution time and maximum absolute error are compared with Gauss-Seidel (GS) and Successive Over-Relaxation (SOR) iterative methods. Based on the performances analysis, the two-point boundary value problems are found to have the most favorable results when solved using CG compared to GS and SOR methods.

Two-point boundary value problems (BVPs) have numerous applications especially in the modeling of most physical phenomena. In this study, the fourth order two solved by using Successive Over cubic non-polynomial spline scheme a numerical experiment was conducted method and the spline scheme in terms of iterations number, execution time and maximum absolute error at different mesh sizes idea, another iterative method was also conducted which is Gauss numerical analysis, the two-point BVPs scheme were found to be best solved by point boundary value problems (BVPs) have numerous applications especially in the modeling of most physical phenomena. In this study, the fourth order two-point BVPs were using Successive Over-Relaxation (SOR) iterative method after discretized with lynomial spline scheme to generate its corresponding sparse linear system. Then, was conducted to determine the performances of the SOR itera the spline scheme in terms of iterations number, execution time and maximum absolute error at different mesh sizes. In order to assess the performances of this proposed idea, another iterative method was also conducted which is Gauss-Seidel (GS).point BVPs together with the cubic non-polynomial spline solved by using the SOR iterative method.

Two-point boundary value problems are commonly used as a numerical test in developing an efficient numerical method. Several researchers studied the application of a cubic non-polynomial spline method to solve the two-point boundary value problems. A preliminary study found that a cubic non-polynomial spline method is better than a standard finite difference method in terms of the accuracy of the solution. Therefore, this paper aims to examine the performance of a cubic non-polynomial spline method through the combination with the full-, half-, and quarter-sweep iterations. The performance was evaluated in terms of the number of iterations, the execution time and the maximum absolute error by varying the iterations from full-, half- to quarter-sweep. A successive over-relaxation iterative method was implemented to solve the large and sparse linear system. The numerical result showed that the newly derived QSSOR method, based on a cubic non-polynomial spline, performed better than the tested FSSOR and HSSOR methods.