New Cubic B-Spline Approximation for Solving Linear Two-Point Boundary-Value Problems

Busyra Latif, Samsul Ariffin Abdul Karim and Ishak Hashim
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Busyra Latif: Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, Cawangan Negeri Sembilan, Kampus Seremban, Seremban 70300, Negeri Sembilan, Malaysia
Samsul Ariffin Abdul Karim: Fundamental and Applied Sciences Department and Centre for Systems Engineering (CSE), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, Seri Iskandar 32610, Perak DR, Malaysia
Ishak Hashim: Department of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, UKM, Bangi 43600, Selangor, Malaysia

Mathematics, 2021, vol. 9, issue 11, 1-13

Abstract: In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O ( h 3 ) accurate. By using this new second-order approximation, the proposed method was O ( h 5 ) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

Keywords: cubic B-spline; two-point boundary value problems; ordinary differential equation; numerical solution; error analysis (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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