Comparative Analysis Between Cubic and Quartic Non-Polynomial Spline Solutions for fourth-order two-Point BVPS

摘要

Fourth-order two-point boundary value problems (BVPs) were solved based on a discretization of cubic and quartic degree of non-polynomial spline general functions. The discretization process began with simple algebraic substitution in order to get the value of all constants and further simplified by replacing certain variables with finite difference method. At the end of this process, the general approximation equations for both degrees of splines were finally developed. To solve the problems iteratively, the single equation of fourth-order problems were first reduced to two separated equations of second-order problems by means of differentiation. Then, the general approximation equations obtained earlier were imposed into these two equations to form two systems of linear equations. After these corresponding linear systems were constructed, Quarter-Sweep Successive-Over Relaxation (QSSOR) method was implemented to solve the two linear systems at varied matrix sizes. In order to assess the performances of this proposed idea, Full-Sweep Successive-Over Relaxation (FSSOR) and Half-Sweep Successive-Over Relaxation (HSSOR) were also conducted. Finally, the performances were presented evidently in terms of iterations number, execution time and maximum absolute error. Based on the performances obtained, the quartic degree of non-polynomial spline was found to be superior compared to the cubic degree when solving the fourth-order two-point BVPs.