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

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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{1}$$\end{document}λ1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _{2}$$\end{document}λ2, 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.


Background
It is well-known that many real life phenomena in physics and engineering can be modelled by systems of linear and nonlinear differential equations. One class of these systems is of second order boundary value problems. The existence of solution to such system was studied in Chen et al. (2005), Cheng and Zhong (2005), Thompson and Tisdell (2002). Consider the following linear system of second-order boundary value problems: where a ≤ x ≤ b, f 1 (x) and f 2 (x) are continuous functions, and a i (x) and b i (x), for i = 1, 2, 3, 4, 5, are real-valued functions of x that are smooth enough.
In 2003, Han and Liu proposed an extension of cubic B-spline of degree four with one free parameter, . This parameter is introduced within the basis function in order to increase the flexibility of the spline curve (Han and Liu 2003). Then, Xu and Wang generalized the extension to degree five and six (Gang and Guo-Zhao 2008). Our goal is to apply the simplest B-spline extension, that is, extended cubic B-spline of degree four, in solving Eq. (1). Linear and singular boundary value problems has already been solved using extended cubic B-spline of degree four and an approach of optimizing has been proposed (Hamid et al. 2011;Goh et al. 2011). The results are promising and thus become the motivation of this study.
In this paper, extended cubic B-spline will be discussed along with the extended cubic B-spline method (ECBM). Optimization of the free parameters and calculations on the truncation error will follow. Three examples will be presented and comparisons with other methods will be made.

Extended cubic B-spline method
Extended cubic B-spline is an extension of B-spline Gang and Guo-Zhao (2008). One free parameter, , is introduced within the basis function where this parameter can be used to alter the shape of the generated curve. The value of can be varied to obtain different numerical results. In this study, this value is optimized to produce approximate solutions with the least error.

Extended cubic B-spline
Suppose that {x i } n i=0 is a uniform partition of a finite interval [a, b] with n ∈ Z + such that a = x 0 < x 1 < · · · < x n = b. The partition can be extended using h = b−a n , x 0 = a, x i = x 0 + ih, and i ∈ Z. Extended cubic B-spline basis function is established from a linear combination of the cubic B-spline basis function (Gang and Guo-Zhao 2008). Here, the blending function of degree four, E 4 i , as shown in (2), is used.
Extended cubic B-spline basis will degenerate into cubic B-spline basis when = 0. For −8 ≤ ≤ 1, cubic B-spline and extended cubic B-spline share the same properties: partition of unity, non-negativity, C 2 continuity, and local suport Hamid (2010). (2) From the basis function, an arbitrary spline curve can be generated by the following formula: As a result, U (x, ) is a piecewise polynomial functions of degree 4. Similarly, for −8 ≤ ≤ 1, cubic B-spline and extended cubic B-spline curves have the same properties: symmetry, geometric invariability, and convex hull Goh et al. (2011). The values of E i and its derivatives E ′ i , E ′′ i at the nodal points are tabulated in Table. 1.

Extended cubic B-spline interpolation
Suppose that the spline curves U (x, 1 ) and V (x, 2 ) are the approximation to the exact solutions, u(x) and v(x), respectively, defined as follows: Therefore, from Table 1, the values and V ′′ (x, 2 ) at knot x i can be simplified into Eqs. (4) and (5). (3) Equations (4) and (5) will be used in simplifying the terms in the system of boundary value problems.

Solution of system of second order boundary value problem
In this part, a collocation approach based on extended cubic B-spline basis functions is used to obtain the numerical solutions of a class of systems of linear second order boundary value problems (1). The approximate solution (3) should satisfy the differential equation at points x i . This can be done by putting (3) into (1) resulting in Eqs. (6)-(9).
Equation (10) can be solved for values of C i 's and D i 's in terms of 1 and 2 by taking Y = X −1 Z. Lastly, the numerical solution can be calculated after obtaining the values of 1 or 2 either by optimization or trial and error (Hamid et al. 2011).
Optimizing the 1 and 2 The approximate analytical solution is of the form where C i 's and D i 's are obtained by solving a linear system of order 2(n + 3) × 2(n + 3). C i 's and D i 's are functions of x, 1 , and 2 . The approach used is adopted from Hamid et al. (2011Hamid et al. ( , 2010. Equation (11) has three free parameters, x, 1 , and 2 . So, U(x) and V(x) can be written as U (x, 1 , 2 ) and V (x, 1 , 2 ) respectively. U (x, 1 , 2 ) and V (x, 1 , 2 ) are piecewise polynomials with n intervals, as in equation (12) and (13). Each U i (x, 1 , 2 ) and V i (x, 1 , 2 ), for i = 1, 2, . . . , n are polynomials of degree four.
From the general form of the problem in (1), f 1 (x) and f 2 (x) are moved to the left-hand side of the equations, as in (14).
Substituting the approximate solutions, U (x, 1 , 2 ) and V (x, 1 , 2 ) and its derivatives into (14), we have Equation (15) is like a version of error formula. From this equation, we have which can be expanded into Eqs. (16) and (17).
Since D 1 (x, 1 , 2 ) and D 2 (x, 1 , 2 ) are piecewise functions with n equations, it is wise to have some representatives from every sub-interval. The representative is taken to be the midpoint of every sub-interval. Therefore, would produce a sequence of 2n expressions containing 1 and 2 , By handling Eqs. (18) and (19) as the error at collocation points, the expressions are combined using the two-norm formula resulting equation (20). This equation measures the accuracy of the approximated solution, U (x, 1 , 2 ) and V (x, 1 , 2 ) without including the exact solution.
Also, from Eq. (20) we can obtain d 2 ( 1 , 2 ) which is assumed to be easier to calculate than the former.
On the other hand, we can combine the expressions using one-norm formula, as in (22).
This is done to make comparisons between results of d 1 ( 1 , 2 ), d 2 ( 1 , 2 ), and d 3 ( 1 , 2 ) in terms of computational time and accuracy. d 3 ( 1 , 2 ) is significantly more simplified that the other two. Finally, we can substitute the optimized value of 1 and 2 in the approximate solution for the problems.

Error estimation
The technique for finding the error estimate as in Kadalbajoo and Kumar (2007) is extended to the system of linear second order differential equations. In this part, a truncation error for the present method in the interval [0, 1] is presented. Suppose that u(x) and v(x) are functions with continuous derivatives in [0, 1]. By using the formulas of u(x) in (4), the following relationship can be obtained.
By using e 1 (x) = U (x, 1 ) − u(x) and substituting relations (27)-(30) in the Taylor series expansion of e 1 (x i + θ h), we obtain Similarly, we can use the definition e 2 (x) = V (x, 2 ) − v(x) to have Therefore, the extended cubic B-spline has a truncation error of order h 2 . Apparently, the value of 1 and 2 have influences on the order.

Results and discussions
Several examples are discussed to demonstrate the efficiency of the proposed method.
The approximate and exact solutions at the nodal points are displayed in Table 3. From the table, the approximate solutions agree with the exact solutions. Hence, for this example, the results are acceptable and accurate. The plots of the numerical results are shown in Figs. 2 and 3. Comparisons between the L ∞ of ECBM, He's homotopy perturbation method (Saadatmandi et al. 2009), and Laplace homotopy analysis method (Ogunlaran and Ademola 2015) were made in Table 4. ECBM produced more accurate results than both methods except for the results of u(x) generated by the Laplace homotopy analysis method (Ogunlaran and Ademola 2015). Moreover, the numerical results for Example

Absolute error
|V ( Tables 5 and 6. In this case, the values of 1 and 2 were obtained by trial and error. From the table, the ECBM produced more accurate results than variational iteration method (VIM) and cubic B-spline method (CBM). The norms for both n are shown in Table 7. It can be observed that ECBM improves the accuracy of CBM significantly.
The approximate and exact solutions at the nodal points are displayed in Table 9. Again, from the table, the approximate solutions agree with the exact solutions. The plots of the    Tables 10 and 11 and compared with reproducing kernel and sinc methods (Geng and Cui 2007;Dehghan and Saadatmandi 2007). The values of 1 and 2 were obtained by trial and error. It can be seen that ECBM produced results with significantly higher accuracy than the other two. The infinite and two norms are shown in Table 12. For this example, ECBM improves the accuracy of CBM for u(x) and gives out similar results for v(x).   Table 13 displays the values of 1 and 2 when d 1 ( 1 , 2 ), d 2 ( 1 , 2 ), and d 3 ( 1 , 2 ) are minimized for n = 5 together with the values of L ∞ and L 2 . Minimizing d 3 ( 1 , 2 ) is the best option because the computational time is significantly less than that of d 1 ( 1 , 2 ) and d 2 ( 1 , 2 ). However, the minimizing values of 1 and 2 are equivalent to CBM. It   can also be observed that minimizing d 2 ( 1 , 2 ) gives similar results with minimizing d 1 ( 1 , 2 ) with a little less computational time.
The approximate and exact solutions at the nodal points are displayed in Table 14. The plots of the numerical results are shown in Figs. 6 and 7. The numerical results for n = 21 and 1 = 2 = 1.25E−14 are shown in Table 15 and compared with CBM Caglar and Caglar (2009). The values of 1 and 2 were obtained by trial and error. It can be seen that ECBM produced slightly more accurate results than CBM. The infinite and two norms are shown in Table 16.

Conclusions
In this research, a new method for finding approximate solutions for a system of second order boundary value problems based on extended cubic B-spline was proposed. This method is called extended cubic B-spline method. The error estimation was carried  Table 14 Comparison of ECBM results with the exact solution for Example 3 when 1 = 0.000000, 2 = 0.000000, and n = 5 x Exact solution u(x)

Absolute error
|V ( 6 Numerical solution U(x) and exact solution u(x) for Example 3 with 1 = 0.000000, 2 = 0.000000, and n = 5 out and the truncation error was found to be of order h 2 , whereby the values of the free parameters 1 and 2 have influence on the order. This method improved the accuracy of its predecessor, CBM, and produced more accurate results than some other numerical methods. It is also found that minimizing the one-norm term, d 3 ( 1 , 2 ) is sufficient to obtain the optimized values of 1 and 2 . More work can be done in the optimizing technique to improve the computational time.