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- Open Access
Extended cubic B-spline method for solving a linear system of second-order boundary value problems
- Ahmed Salem Heilat^{1}Email author,
- Nur Nadiah Abd Hamid^{1} and
- Ahmad Izani Md. Ismail^{1}
- Received: 9 May 2016
- Accepted: 27 July 2016
- Published: 9 August 2016
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, \(\lambda _{1}\) and \(\lambda _{2}\), 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.
Keywords
- Boundary value problem
- System
- Linear
- Extended cubic B-spline
Background
There are many studies on the solutions of linear and nonlinear systems of second-order boundary value problems approximately. Amongst others are variational iteration, reproducing kernel, sinc-collocation, modified homotopy analysis, continuous genetic algorithm, He’s homotopy perturbation, Laplace homotopy analysis, homotopy perturbation-reproducing kernel, and local radial basis function based differential quadrature methods (Lu 2007; Geng and Cui 2007; Dehghan and Saadatmandi 2007; Bataineh et al. 2009; Arqub and Abo-Hammour 2014; Saadatmandi et al. 2009; Ogunlaran and Ademola 2015; Geng and Cui 2011; Dehghan and Nikpour 2013). The main purpose of our present study is to apply a spline function in solving Eq. (1). This equation had already been treated using cubic B-spline, cubic B-spline scaling functions, sinc-collocation, and spline collocation approaches (Caglar and Caglar 2009; Dehghan and Lakestani 2008; El-Gamel 2012; Khuri and Sayfy 2009).
In 2003, Han and Liu proposed an extension of cubic B-spline of degree four with one free parameter, \(\lambda\). 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 \(\lambda\) 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, \(\lambda\), is introduced within the basis function where this parameter can be used to alter the shape of the generated curve. The value of \(\lambda\) 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
Coefficient of \(E_{i}\), \(E_{i}^{\prime}\), and \(E_{i}^{\prime\prime}\)
x | \(x_{i}\) | \(x_{i+1}\) | \(x_{i+2}\) | \(x_{i+3}\) | \(x_{i+4}\) |
---|---|---|---|---|---|
\(E_{i}\) | 0 | \(\frac{4-\lambda }{24}\) | \(\frac{8+\lambda }{12}\) | \(\frac{4-\lambda }{24}\) | 0 |
\(E_{i}^{\prime}\) | 0 | \(\frac{-1}{2h}\) | \(\frac{0}{h}\) | \(\frac{1}{2h}\) | 0 |
\(E_{i}^{\prime\prime}\) | 0 | \(\frac{2+\lambda }{2h^{2}}\) | \(\frac{-2-\lambda }{h^{2}}\) | \(\frac{2+\lambda }{2h^{2}}\) | 0 |
Extended cubic B-spline interpolation
Solution of system of second order boundary value problem
Optimizing the \(\lambda _{1}\) and \(\lambda _{2}\)
Error estimation
Results and discussions
Example 1
Table 2 displays the values of \(\lambda _{1}\) and \(\lambda _{2}\) when \(d_{1}(\lambda _{1},\lambda _{2})\), \(d_{2}(\lambda _{1},\lambda _{2})\), and \(d_{3}(\lambda _{1},\lambda _{2})\) are minimized for \(n=5\). The \(L_{\infty }\) and \(L_{2}\) for each pair are also presented. From the table, it can be deduced that minimizing \(d_{3}(\lambda _{1},\lambda _{2})\) is the best option because the results are comparable and the computational time is significantly less than that of \(d_{1}(\lambda _{1},\lambda _{2})\) and \(d_{2}(\lambda _{1},\lambda _{2})\). Therefore, the chosen values of \(\lambda _{1}\) and \(\lambda _{2}\) are −6.639145E−02 and \(1.161882E{-}06\), respectively. Also, it can be observed that minimizing \(d_{2}(\lambda _{1},\lambda _{2})\) gives similar results with minimizing \(d_{1}(\lambda _{1},\lambda _{2})\) with significantly less computational time.
Computational time and norms for different optimization equations \(d_{1}(\lambda _{1},\lambda _{2})\), \(d_{2}(\lambda _{1},\lambda _{2})\), and \(d_{3}(\lambda _{1},\lambda _{2})\) with \(n=5\)
Minimization values | \(d_{1}(\lambda _{1},\lambda _{2})\) | \(d_{2}(\lambda _{1},\lambda _{2})\) | \(d_{3}(\lambda _{1},\lambda _{2})\) |
---|---|---|---|
\(\lambda _{1}\) | −6.639979E−02 | −6.639979E−02 | −6.639145E−02 |
\(\lambda _{2}\) | −1.230437E−06 | −1.230522E−06 | 1.161882E−06 |
Computational time (s) | 1.306340E+04 | 2.728410E+03 | 2.230830E+00 |
\(L_{\infty }\) | 1.377934E−04 | 1.377934E−04 | 1.413576E−04 |
\(L_{2}\) | 2.306527E−04 | 2.306527E−04 | 2.364995E−04 |
Comparison of ECBM results with the exact solution for Example 1 when \(\lambda _{1}=-6.639145E{-}02\), \(\lambda _{2}=1.161882E{-}06\), and \(n=5\)
x | Exact solution u(x) | Approx. solution U(x) | Absolute error \(|U(x)-u(x)|\) | Exact solution v(x) | Approx. solution V(x) | Absolute error \(|V(x)-v(x)|\) |
---|---|---|---|---|---|---|
0.2 | 0.587785 | 0.587696 | 8.897274E−05 | −0.160000 | −0.160004 | 3.641560E−06 |
0.4 | 0.951057 | 0.950915 | 1.413501E−04 | −0.240000 | −0.240006 | 6.478141E−06 |
0.6 | 0.951057 | 0.950915 | 1.413576E−04 | −0.240000 | −0.240007 | 7.169404E−06 |
0.8 | 0.587785 | 0.587696 | 8.891932E−05 | −0.160000 | −0.160005 | 4.793718E−06 |
He’s homotopy perturbation method | Laplace homotopy analysis method | ECBM (\(\lambda _{1}=-6.639145E{-}02\), \(\lambda _{2}=1.161882E{-}06\)) | |
---|---|---|---|
U (x) | 2.1E−04 | 2.2E−05 | 1.4E−04 |
V (x) | 3.2E−04 | 1.1E−05 | 7.2E−06 |
x | VIM | CBM | ECBM (\(\lambda _{1}=\lambda _{2}=0\)) | ECBM (\(\lambda _{1}=-1.0E{-}03, \lambda _{2}=0\)) |
---|---|---|---|---|
0.1 | 3.30E−04 | 1.40E−04 | 1.30E−04 | 2.83E−06 |
0.2 | 2.51E−03 | 2.80E−04 | 2.56E−04 | 5.55E−06 |
0.3 | 7.84E−03 | 3.90E−04 | 3.60E−04 | 7.81E−06 |
0.4 | 1.66E−02 | 4.60E−04 | 4.28E−04 | 9.30E−06 |
0.5 | 2.77E−02 | 4.80E−04 | 4.52E−04 | 9.82E−06 |
0.6 | 3.87E−02 | 4.60E−04 | 4.28E−04 | 9.30E−06 |
0.7 | 4.59E−02 | 3.90E−04 | 3.60E−04 | 7.81E−06 |
0.8 | 4.49E−02 | 2.80E−04 | 2.56E−04 | 5.56E−06 |
0.9 | 3.09E−02 | 1.50E−04 | 1.30E−04 | 2.83E−06 |
Absolute errors of CBM Caglar and Caglar (2009) and ECBM results for Example 1 with \(n=41\) for v(x)
x | CBM | ECBM (\(\lambda _{1}=\lambda _{2}=0\)) | ECBM (\(\lambda _{1}=-1.0E{-}03, \lambda _{2}=0\)) |
---|---|---|---|
0.1 | 5.74E−06 | 5.74E−06 | 1.25E−07 |
0.2 | 1.13E−05 | 1.13E−05 | 2.46E−07 |
0.3 | 1.64E−05 | 1.64E−05 | 3.56E−07 |
0.4 | 2.03E−05 | 2.03E−05 | 4.42E−07 |
0.5 | 2.26E−05 | 2.26E−05 | 4.91E−07 |
0.6 | 2.26E−05 | 2.26E−05 | 4.92E−07 |
0.7 | 2.01E−05 | 2.01E−05 | 4.37E−07 |
0.8 | 1.51E−05 | 1.51E−05 | 3.29E−07 |
0.9 | 8.14E−06 | 8.14E−06 | 1.76E−07 |
\(L_{\infty }\) and \(L_{2}\) of ECBM results for Example 1
n | 5 | 5 | 41 | 41 |
---|---|---|---|---|
\(\lambda _{1}\) | 0.000000 | −6.639145E−02 | 0.000000 | −1.000000E−03 |
\(\lambda _{2}\) | 0.000000 | 1.161882E−06 | 0.000000 | 0.000000 |
\(L_{\infty }\) of U(x) | 2.791929E−02 | 1.413576E−04 | 4.518529E−04 | 9.817274E−06 |
\(L_{\infty }\) of V(x) | 1.423849E−03 | 7.169404E−06 | 2.263578E−05 | 4.917602E−07 |
\(L_{2}\) of U(x) | 4.600584E−02 | 2.362253E−04 | 9.969665E−04 | 2.165970E−05 |
\(L_{2}\) of V(x) | 2.262625E−03 | 1.138452E−05 | 5.066609E−05 | 1.100638E−06 |
Example 2
Table 8 displays the values of \(\lambda _{1}\) and \(\lambda _{2}\) when \(d_{1}(\lambda _{1},\lambda _{2})\), \(d_{2}(\lambda _{1},\lambda _{2})\), and \(d_{3}(\lambda _{1},\lambda _{2})\) are minimized for \(n=5\), with their respective \(L_{\infty }\) and \(L_{2}\). Again, minimizing \(d_{3}(\lambda _{1},\lambda _{2})\) is the best option because the results are comparable and the computational time is significantly less than that of \(d_{1}(\lambda _{1},\lambda _{2})\) and \(d_{2}(\lambda _{1},\lambda _{2})\). Therefore, the chosen values of \(\lambda _{1}\) and \(\lambda _{2}\) are −1.269208E−02 and \(-6.634523E{-}02\), respectively. For this example, minimizing \(d_{2}(\lambda _{1},\lambda _{2})\) gives similar results with minimizing \(d_{1}(\lambda _{1},\lambda _{2})\) with almost similar computational time.
Computational time and norms for different optimization equations \(d_{1}(\lambda _{1},\lambda _{2})\), \(d_{2}(\lambda _{1},\lambda _{2})\), and \(d_{3}(\lambda _{1},\lambda _{2})\) with \(n=5\)
Minimization values | \(d_{1}(\lambda _{1},\lambda _{2})\) | \(d_{2}(\lambda _{1},\lambda _{2})\) | \(d_{3}(\lambda _{1},\lambda _{2})\) |
---|---|---|---|
\(\lambda _{1}\) | −1.273122E−02 | −1.273121E−02 | −1.269208E−02 |
\(\lambda _{2}\) | −6.634562E−02 | −6.634562E−02 | −6.634523E−02 |
Computational time (s) | 5.517106E+02 | 5.196057E+02 | 2.959325E+01 |
\(L_{\infty }\) | 1.750978E−04 | 1.750978E−04 | 1.750618E−04 |
\(L_{2}\) | 2.913261E−04 | 2.913260E−04 | 2.926986E−04 |
Comparison of ECBM results with the exact solution for Example 2 when \(\lambda _{1}=-0.012692\), \(\lambda _{2}=-0.066345\), and \(n=5\)
x | Exact solution u(x) | Approx. solution U(x) | Absolute error \(|U(x)-u(x)|\) | Exact solution v(x) | Approx. solution V(x) | Absolute error \(|V(x)-v(x)|\) |
---|---|---|---|---|---|---|
0.2 | 0.317871 | 0.317853 | 1.769288E−05 | 0.587785 | 0.587676 | 1.093618E−04 |
0.4 | 0.467302 | 0.467284 | 1.800318E−05 | 0.951057 | 0.950881 | 1.750618E−04 |
0.6 | 0.451714 | 0.451696 | 1.804713E−05 | 0.951057 | 0.950882 | 1.744319E−04 |
0.8 | 0.286942 | 0.286926 | 1.603373E−05 | 0.587785 | 0.587678 | 1.068617E−04 |
x | Reproducing kernel | Sinc method | ECBM (\(\lambda _{1}=\lambda _{2}=0\)) | ECBM (\(\lambda _{1}=\lambda _{2}=-1.0E{-}03\)) |
---|---|---|---|---|
0.08 | 3.3E−03 | 3.2E−03 | 1.3E−04 | 1.4E−05 |
0.24 | 7.7E−03 | 9.4E−04 | 2.7E−04 | 1.1E−05 |
0.40 | 9.7E−03 | 2.0E−03 | 2.7E−04 | 2.1E−05 |
0.56 | 9.5E−03 | 2.2E−04 | 2.0E−04 | 5.9E−05 |
0.72 | 7.3E−03 | 4.1E−03 | 9.4E−05 | 7.8E−05 |
0.88 | 3.4E−03 | 1.0E−02 | 1.6E−05 | 5.6E−05 |
0.96 | 1.1E−03 | 2.1E−03 | 3.6E−08 | 2.3E−05 |
x | Reproducing kernel | Sinc method | ECBM (\(\lambda _{1}=\lambda _{2}=0\)) | ECBM (\(\lambda _{1}=\lambda _{2}=-1.0E{-}03\)) |
---|---|---|---|---|
0.08 | 7.7E−03 | 1.5E−03 | 3.8E−04 | 2.2E−04 |
0.24 | 2.2E−02 | 7.0E−03 | 9.9E−04 | 6.0E−04 |
0.40 | 2.7E−02 | 7.4E−03 | 1.3E−03 | 8.3E−04 |
0.56 | 2.7E−02 | 1.0E−02 | 1.4E−03 | 8.6E−04 |
0.72 | 2.0E−02 | 4.4E−03 | 1.1E−03 | 6.8E−04 |
0.88 | 9.4E−03 | 2.1E−02 | 5.0E−04 | 3.3E−04 |
0.96 | 3.1E−03 | 6.9E−03 | 1.7E−04 | 1.1E−04 |
\(L_{\infty }\) and \(L_{2}\) of ECBM results for Example 2
n | 5 | 5 | 25 | 25 |
---|---|---|---|---|
\(\lambda _{1}\) | 0.000000 | −1.269208E−02 | 0.000000 | −1.000000E−03 |
\(\lambda _{2}\) | 0.000000 | −6.634523E−02 | 0.000000 | −1.000000E−03 |
\(L_{\infty }~ {\rm{of}}~ U(x)\) | 2.086834E−03 | 1.804713E−05 | 2.720423E−04 | 7.798961E−05 |
\(L_{\infty }~ {\rm{of}}~ V(x)\) | 1.750618E−04 | 1.750618E−04 | 1.364287E−03 | 8.604698E−04 |
\(L_{2 }~ {\rm{of}}~ U(x)\) | 2.087051E−03 | 3.492752E−05 | 4.590374E−04 | 1.179224E−04 |
\(L_{2 }~{\rm{of}}~V(x)\) | 2.906072E−04 | 2.906072E−04 | 2.491362E−03 | 1.556034E−03 |
Example 3
Table 13 displays the values of \(\lambda _{1}\) and \(\lambda _{2}\) when \(d_{1}(\lambda _{1},\lambda _{2})\), \(d_{2}(\lambda _{1},\lambda _{2})\), and \(d_{3}(\lambda _{1},\lambda _{2})\) are minimized for \(n=5\) together with the values of \(L_{\infty }\) and \(L_{2}\). Minimizing \(d_{3}(\lambda _{1},\lambda _{2})\) is the best option because the computational time is significantly less than that of \(d_{1}(\lambda _{1},\lambda _{2})\) and \(d_{2}(\lambda _{1},\lambda _{2})\). However, the minimizing values of \(\lambda _{1}\) and \(\lambda _{2}\) are equivalent to CBM. It can also be observed that minimizing \(d_{2}(\lambda _{1},\lambda _{2})\) gives similar results with minimizing \(d_{1}(\lambda _{1},\lambda _{2})\) with a little less computational time.
Computational time and norms for different optimization equations \(d_{1}(\lambda _{1},\lambda _{2})\), \(d_{2}(\lambda _{1},\lambda _{2})\), and \(d_{3}(\lambda _{1},\lambda _{2})\) with \(n=5\)
Minimization values | \(d_{1}(\lambda _{1},\lambda _{2})\) | \(d_{2}(\lambda _{1},\lambda _{2})\) | \(d_{3}(\lambda _{1},\lambda _{2})\) |
---|---|---|---|
\(\lambda _{1}\) | 0.000000 | 0.000000 | 0.000000 |
\(\lambda _{2}\) | 0.000000 | 0.000000 | 0.000000 |
Computational time (s) | 7.314018E+01 | 6.738385E+01 | 4.973284E+00 |
\(L_{\infty }\) | 3.691492E−15 | 3.691492E−15 | 3.691492E−15 |
\(L_{2}\) | 6.058413E−15 | 6.058413E−15 | 6.058413E−15 |
Comparison of ECBM results with the exact solution for Example 3 when \(\lambda _{1}=0.000000\), \(\lambda _{2}=0.000000\), and \(n=5\)
x | Exact solution u(x) | Approx. solution U(x) | Absolute error \(|U(x)-u(x)|\) | Exact solution v(x) | Approx. solution V(x) | Absolute error \(|V(x)-v(x)|\) |
---|---|---|---|---|---|---|
0.2 | −0.160000 | −0.160000 | 4.163336E−16 | 0.160000 | 0.160000 | 4.718448E−16 |
0.4 | −0.240000 | −0.240000 | 2.775558E−17 | 0.240000 | 0.240000 | 6.106227E−16 |
0.6 | −0.240000 | −0.240000 | 9.992007E−16 | 0.240000 | 0.240000 | 2.775558E−16 |
0.8 | −0.160000 | −0.160000 | 3.469447E−15 | 0.160000 | 0.160000 | 3.691492E−15 |
Comparison of norms of CBM and ECBM for Example 3 when \(n=21\) for u(x) and v(x)
Errors | ECBM (\(\lambda _{1}=\lambda _{2}=0\)) | ECBM (\(\lambda _{1}=\lambda _{2}=1.25E{-}14\) ) | ||
---|---|---|---|---|
u(x) | v(x) | u(x) | v(x) | |
\(L_{\infty }\) | 3.720357E−13 | 2.531308E−13 | 1.725009E−13 | 1.668943E−13 |
L2 | 4.367056E−13 | 4.365110E−13 | 2.930975E−13 | 2.223093E−13 |
\(L_{\infty }\) and \(L_{2}\) of ECBM results for Example 3
n | 5 | 21 | 21 |
---|---|---|---|
\(\lambda _{1}\) | 0.000000 | 0.000000 | \(1.250000E{-}14\) |
\(\lambda _{2}\) | 0.000000 | 0.000000 | \(1.250000E{-}14\) |
\(L_{\infty }\) of U(x) | 3.469447E−15 | 3.720357E−13 | 1.725009E−13 |
\(L_{\infty }~{\rm{of}}~V(x)\) | 3.691492E−15 | 2.530308E−13 | 1.668943E−13 |
\(L_{2 }~{\rm{of}}~U(x)\) | 3.634497E−15 | 4.367056E−13 | 2.930975E−13 |
\(L_{2 }~{\rm{of}}~V(x)\) | 3.781487E−15 | 4.365110E−13 | 2.223093E−13 |
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 out and the truncation error was found to be of order \(h^2\), whereby the values of the free parameters \(\lambda _{1}\) and \(\lambda _{2}\) 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, \(d_3(\lambda _1,\lambda _2)\) is sufficient to obtain the optimized values of \(\lambda _1\) and \(\lambda _2\). More work can be done in the optimizing technique to improve the computational time.
Declarations
Authors' contributions
ASH suggested the method and the problem and wrote the first version of the paper and carried out the works to generate results using Mathematica. NNAH checked the paper and the Mathematica program, gave constructive comments and suggestions to ASH to improve the quality of the paper. AIMI did the final checking and reviewing. All authors read and approved the final manuscript.
Acknowlegements
We thank Dr. Abedel-Karrem Alomari and Maher Hailat for their help in this study. Also the authors would like to thank the Editor-in-Chief and the reviewers for their valuable suggestions to improve this work.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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