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(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) 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.
Highlights
Boundary value problems (BVPs) have been extensively investigated in the fields of physics, chemistry and engineering
Numerous methods have been implemented throughout the years to approximate the solutions of linear and nonlinear two-point BVPs, such as the variational approach, finite difference (FDM), finite element (FEM), finite volume (FVM) and shooting (LSM) [1,2,3]
In 2006, Caglar et al [11] replaced the cubic spline with a cubic B-spline basis function to solve two-point BVPs and named it the cubic B-spline interpolation method (BSI)
Summary
Boundary value problems (BVPs) have been extensively investigated in the fields of physics, chemistry and engineering. Numerous methods have been implemented throughout the years to approximate the solutions of linear and nonlinear two-point BVPs, such as the variational approach, finite difference (FDM), finite element (FEM), finite volume (FVM) and shooting (LSM) [1,2,3]. In 2006, Caglar et al [11] replaced the cubic spline with a cubic B-spline basis function to solve two-point BVPs and named it the cubic B-spline interpolation method (BSI). Several cubic B-spline based numerical approaches have been widely applied to solve linear and nonlinear BVPs [12,13,14,15]. The extended cubic B-spline and cubic trigonometric B-spline were studied by Hamid et al [16,17] as solutions to linear two-point BVPs. It was found that the cubic trigonometric B-spline provided better results compared with the cubic BSI
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