Quartic non-polynomial spline solution for solving two-point boundary value problems by using Conjugate Gradient Iterative Method

Abstract

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.

Solution of fourth-order two-point BVPs with cubic non-polynomial spline and SOR iterative method

“…This indicates that the non-polynomial spline scheme together with the SOR method can cope with the accumulated round-off error better than GS method. Thus, For future work, it is highly recommended to extend this study for solving high order two-point BVPs with different degree of spline such as quartic, quintic and so on by referring to [1][2][3][4][5], [14][15]22]. …”

“…In response to the vast applications of two-point BVPs as stated by [16][17][18][19][20] regardless of fields, it is the main focus of this present study to initiate a numerical idea in term of technique and algorithm, as well as to provide a reference for future research. To be precise, a f spline scheme at a different degree for solving two order with different iterative methods.…”

“…1 of each subinterval is given as follows Then, Equation (3) and (5) were solved iteratively at once by using SOR iter referring to [12][13][14][15]. In order to assess the performance of this proposed idea clearly in term of iterations number, execution time and maximum absolute error, GS iterative method was also conducted for comparison purpose.…”

“…By imposing Equation (16) into Equation (15), it yields to the SOR iteration equation in the following form…”

Non-polynomial spline finite difference method for solving second order boundary value problem

“…consider the linear second order BVP of the form [7,10]   ( ) ( ) ( ) ( ) , , y x r x y s x y g x x a b       (1) subject to the boundary conditions (B.Cs) Each non-polynomial spline segment has the form [4] ( ) ( ) sinh( ( )) …”