Extended cubic B-spline method for solving a linear system of second-order boundary value problems

Abstract

A method based on extended cubic B-spline is proposed to solve a linear system of second-order boundary value problems. In this method, two free parameters, and , play an important role in producing accurate results. Optimization of these parameters are carried out and the truncation error is calculated. This method is tested on three examples. The examples suggest that this method produces comparable or more accurate results than cubic B-spline and some other methods.

Extended cubic B-spline method for solving a system of non-linear second-order boundary value problems

“…Lastly, the values of λ 1 and λ 2 can be determined by optimizing the one norm as discussed in [17]. Hence, the approximate analytical solution can be generated.…”

“…The truncation error for the system in (2.4) has already been calculated in [17] to be of order h 2 . The errors are also dependent on the values of λ 1 and λ 2 .…”

“…Extended cubic B-spline method (ECBSM) was developed to solve linear and singular two point boundary value problems involving ordinary differential equations in [1,15]. The same approach was also used to solve a linear system of second order boundary value problems by the present authors [17]. Two free parameters were optimized to obtain accurate results.…”

New Spline Method for Solving Linear Two-Point Boundary Value Problems

“…This equation had already been treated using cubic B-spline method (CBSM), trigonometric cubic B-spline method (TCBSM), extended cubic B-spline method (ECBSM), and BS2 methods. ECBSM is by far the efficient spline-based method in producing accurate results [15][16][17][18]. Cubic spline is used to obtain the solution at any point in the range [10][11][12][13].…”

“…The ECBSM have a free parameter λ, and this parameter is important to give more accurate results. The value of λ can be obtained by optimization [17]. BS2 methods give more accurate solution [18].…”