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Midknot cubic nonpolynomial spline for a system of secondorder boundary value problems
 Qinxu Ding^{1} and
 Patricia J. Y. Wong^{1}Email authorView ORCID ID profile
 Received: 4 January 2018
 Accepted: 24 September 2018
 Published: 11 October 2018
Abstract
In this paper, a midknot cubic nonpolynomial spline is applied to obtain the numerical solution of a system of secondorder boundary value problems. The numerical method is proved to be uniquely solvable and it is of secondorder accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature.
Keywords
 Cubic nonpolynomial spline
 Secondorder
 Boundary value problem
 Numerical solution
1 Introduction
There are substantial interests on the numerical treatment of the problem (1.1). Noor and Khalifa [19] have used a collocation method with cubic Bsplines as basis functions to solve (1.1), while the wellknown Numerov method and finite difference schemes based on the central difference have been employed in [22]. Thereafter, AlSaid et al. [5] show that cubic spline method gives numerical solutions that are more accurate than that computed by quintic spline and finite difference techniques. The numerical results of [5, 19, 22] indicate firstorder accuracy for these methods. In [4], a twostage method is developed where a finite difference scheme is first employed to obtain the numerical solutions at midknots of a uniform mesh, then a secondorder interpolation is used to obtain the numerical solutions at the knots. This method is of secondorder accuracy. Other proven secondorder accurate methods include polynomial spline methods that employ quadratic spline [1], cubic spline [2, 3] and quintic spline [6]. The numerical solutions are obtained at midknots of a uniform mesh in [1–3], while numerical solutions are obtained at the knots in [6]. These polynomial spline methods use ‘continuous’ spline, and derivatives of the spline are involved in the spline relations. On the other hand, discrete spline uses differences instead of derivatives in the spline relations. In [8], Chen and Wong have developed a deficient discrete cubic spline method for (1.1). It is proved that the accuracy of the method is two, and the numerical experiments demonstrate better accuracy over polynomial spline methods.
Besides continuous polynomial splines, nonpolynomial splines have also been applied to solve (1.1). Nonpolynomial spline, also known as parametric spline [13], depends on a parameter \(k>0\), and reduces to the ordinary cubic or quintic spline when \(k\to0\). Due to the parameter k, the numerical solutions obtained by nonpolynomial splines in the literature are observed to be more accurate than that computed by polynomial splines. In fact, a cubic nonpolynomial spline method has been proposed by Khan and Aziz [12] and subsequently by SirajulIslam and Tirmizi [25] to solve (1.1) at the knots of a uniform mesh. The method is shown to be of order two, and numerical results indicate better accuracy over polynomial spline methods. Higher degree nonpolynomial splines have also been used in higherorder boundary value problems, for example quartic nonpolynomial spline for thirdorder boundary value problem [24, 26], quintic nonpolynomial spline for fourthorder boundary value problem [14] and sextic nonpolynomial spline for fifthorder boundary value problem [15]. Out of all these work, only [26] gives the numerical solutions of the thirdorder boundary value problem at midknots of a uniform mesh while the rest obtains the numerical solutions at the knots. The methods mentioned so far yield discrete numerical schemes. There are also iterative methods such as Adomian decomposition method [18] and variational iteration method [20]. Both of these methods do not require discretization.
Motivated by the above work especially those involving the use of nonpolynomial splines, in this paper we shall develop a cubic nonpolynomial spline scheme at midknots of a uniform mesh for the problem (1.1). The unique solvability and convergence analysis will be carried out which indicates a secondorder accurate method. Finally, three examples will be presented to illustrate the numerical efficiency and the better performance over other methods in the literature.
2 Midknot cubic nonpolynomial spline method
Throughout the paper, for any function \(v(x)\) we shall denote \(v^{(j)}(x_{i})=v^{(j)}_{i}\) and likewise \(v^{(j)}(x_{i1/2})=v^{(j)}_{i1/2}\). In the following, we define the cubic nonpolynomial spline in terms of midknots of the mesh Ω. Note that [13] gives a similar definition but in terms of the knots of Ω.
Definition 2.1
Remark 2.1
When \(k\to0\), we have \((\alpha,\beta)\to (\frac{1}{6},\frac{1}{3} )\) and the cubic nonpolynomial spline relation of Eq. (2.5) reduces to the wellknown cubic spline relation. Further, for the consistency of relation (2.5), we have \(2\alpha+2\beta=1\) [13].
Remark 2.2
Due to the consistency relation \(2\alpha+2\beta=1\), (2.8) immediately gives \(t_{i}=O(h^{4}),~2\leq i\leq n1\). If, in addition, \(\alpha=\frac{1}{12}\) (which implies \(\beta=\frac{5}{12}\)), then (2.8) yields \(t_{i}=O(h^{6}),~2\leq i\leq n1\).

for \(2\leq i \leq\frac{n}{4}1\),$$ S_{i3/2}2S_{i1/2}+S_{i+1/2}=h^{2} (\alpha f_{i3/2}+2\beta f_{i1/2}+\alpha f_{i+1/2} ); $$(2.10)

for \(i=\frac{n}{4}\),$$\begin{aligned} &S_{n/43/2}2S_{n/41/2}+ \bigl(1\alpha h^{2} g_{n/4+1/2} \bigr)S_{n/4+1/2} \\ &\quad =h^{2} \bigl[\alpha f_{n/43/2}+2\beta f_{n/41/2}+ \alpha (f_{n/4+1/2}+r ) \bigr]; \end{aligned}$$(2.11)

for \(i=\frac{n}{4}+1\),$$\begin{aligned} &S_{n/41/2}+ \bigl(22\beta h^{2}g_{n/4+1/2} \bigr)S_{n/4+1/2}+ \bigl(1\alpha h^{2}g_{n/4+3/2} \bigr)S_{n/4+3/2} \\ &\quad =h^{2} \bigl[\alpha f_{n/41/2}+2\beta (f_{n/4+1/2}+r )+\alpha (f_{n/4+3/2}+r ) \bigr]; \end{aligned}$$(2.12)

for \(\frac{n}{4}+2\leq i \leq\frac{3n}{4}1\),$$\begin{aligned} & \bigl(1\alpha h^{2}g_{i3/2} \bigr)S_{i3/2}+ \bigl(22\beta h^{2}g_{i1/2} \bigr)S_{i1/2}+ \bigl(1\alpha h^{2}g_{i+1/2} \bigr)S_{i+1/2} \\ &\quad =h^{2} \bigl[\alpha (f_{i3/2}+r )+2\beta (f_{i1/2}+r )+\alpha (f_{i+1/2}+r ) \bigr]; \end{aligned}$$(2.13)

for \(i=\frac{3n}{4}\),$$\begin{aligned} & \bigl(1\alpha h^{2}g_{3n/43/2} \bigr)S_{3n/43/2}+ \bigl(22\beta h^{2}g_{3n/41/2} \bigr)S_{3n/41/2}+S_{3n/4+1/2} \\ &\quad =h^{2} \bigl[\alpha (f_{3n/43/2}+r )+2\beta (f_{3n/41/2}+r )+\alpha f_{3n/4+1/2} \bigr]; \end{aligned}$$(2.14)

for \(i=\frac{3n}{4}+1\),$$\begin{aligned} & \bigl(1\alpha h^{2}g_{3n/41/2} \bigr)S_{3n/41/2}2S_{3n/4+1/2}+S_{3n/4+3/2} \\ &\quad =h^{2} \bigl[\alpha (f_{3n/41/2}+r )+2\beta f_{3n/4+1/2}+\alpha f_{3n/4+3/2} \bigr]; \end{aligned}$$(2.15)

for \(\frac{3n}{4}+2\leq i \leq n1\),$$ S_{i3/2}2S_{i1/2}+S_{i+1/2}=h^{2} (\alpha f_{i3/2}+2\beta f_{i1/2}+\alpha f_{i+1/2} ). $$(2.16)
3 Solvability and convergence
Remark 3.1
Lemma 3.1
([27])
Lemma 3.2
([11])
We are now ready to establish the unique solvability and the convergence of the midknot cubic nonpolynomial spline scheme in the following theorem.
Theorem 3.1
Proof
On the other hand, for the special case \(\alpha=\frac{1}{12}\), \(\beta=\frac{5}{12}\), we have from (3.9) that \(\T\=O(h^{6})\). So by using a similar argument as above, we obtain \(\E\\leq O(h^{4})\), which indicates that (3.1) is a fourthorder convergence method. However, the solution of problem (1.1) exists continuously only up to the second derivative. Therefore, the numerical method is only secondorder accurate over the whole interval for the special case \(\alpha=\frac{1}{12}\), \(\beta=\frac{5}{12}\). Indeed, a similar conclusion can be observed in [4–6, 8, 12, 25]. In summary, the numerical method (3.1) is of second order for all α and β satisfying \(2\alpha+2\beta=1\). □
Remark 3.2
4 Application to obstacle boundary value problem
In Example 4.1, we shall consider a wellknown special case of the system (4.6) when \(f=0\). This special case is first discussed in [19] and subsequently considered in almost every paper on system of secondorder boundary value problems.
Example 4.1
([19])
(Example 4.1) Maximum absolute errors
Methods  h = π/20  h = π/40  h = π/80 

Midknot cubic nonpolynomial spline α = 1/8, β = 3/8  2.40e − 04  6.34e − 05  1.63e − 05 
Midknot cubic nonpolynomial spline α = 1/10, β = 2/5  3.44e − 04  9.11e − 05  2.34e − 05 
Midknot cubic nonpolynomial spline α = 1/12, β = 5/12  4.14e − 04  1.09e − 04  2.81e − 05 
Midknot cubic nonpolynomial spline α = 1/14, β = 3/7  4.64e − 04  1.23e − 04  3.15e − 05 
Midknot cubic nonpolynomial spline α = 1/16, β = 7/16  5.01e − 04  1.33e − 04  3.40e − 05 
Parametric cubic spline [12] α = 1/8, β = 3/8  8.62e − 04  2.47e − 04  6.57e − 05 
Parametric cubic spline [12] α = 1/10, β = 2/5  7.74e − 04  2.21e − 04  5.89e − 05 
Parametric cubic spline [12] α = 1/12, β = 5/12  7.16e − 04  2.04e − 04  5.43e − 05 
Parametric cubic spline [12] α = 1/14, β = 3/7  6.74e − 04  1.92e − 04  5.11e − 05 
6.43e − 04  1.83e − 04  4.87e − 05  
Deficient discrete cubic spline [8]  1.19e − 03  3.04e − 04  7.68e − 05 
Cubic spline [3]  1.26e − 03  3.29e − 04  8.43e − 05 
Modified Numerov method [4]  1.65e − 03  4.33e − 04  1.11e − 04 
Cubic spline [2]  1.94e − 03  4.99e − 04  1.27e − 04 
Quadratic spline [1]  2.20e − 03  5.87e − 04  1.51e − 04 
Quintic spline [6]  2.57e − 03  7.31e − 04  1.94e − 04 
Collocationcubic B spline [19]  1.40e − 02  7.71e − 03  4.04e − 03 
Cubic spline [5]  1.80e − 02  9.13e − 03  4.60e − 03 
Quintic spline [5]  1.82e − 02  9.17e − 03  4.61e − 03 
Numerov [22]  2.32e − 02  1.21e − 02  6.17e − 03 
Finite difference scheme [22]  2.50e − 02  1.29e − 02  6.58e − 03 
From Table 1, the numerical results confirm that our method is of second order. Compared to the parametric cubic spline method [12, 25], our method gives the smallest errors for all cases of \((\alpha,\beta)\). Furthermore, our method outperforms all other methods [1–6, 8, 19, 22] in all cases.
In the next two examples, unlike Example 4.1, we consider the problem (1.1) with nonzero f.
Example 4.2
(Example 4.2) Maximum absolute errors and convergence orders
h  MCNS  MCNS  MCNS  MCNS  

α = 1/16  β = 7/16  α = 1/14  β = 3/7  α = 1/12  β = 5/12  α = 1/10  β = 2/5  
∥E∥  order  ∥E∥  order  ∥E∥  order  ∥E∥  order  
\(\frac{\pi}{20}\)  7.10e − 03  7.19e − 03  7.30e − 03  7.46e − 03  
\(\frac{\pi}{40}\)  1.86e − 03  1.93  1.88e − 03  1.94  1.91e − 03  1.93  1.96e − 03  1.93 
\(\frac{\pi}{80}\)  4.76e − 04  1.97  4.82e − 04  1.96  4.90e − 04  1.96  5.02e − 04  1.97 
\(\frac{\pi}{160}\)  1.21e − 04  1.98  1.22e − 04  1.98  1.24e − 04  1.98  1.27e − 04  1.98 
\(\frac{\pi}{320}\)  3.03e − 05  2.00  3.07e − 05  1.99  3.12e − 05  1.99  3.19e − 05  1.99 
h  MCNS  DDCS  PCS  PCS  

α = 1/8  β = 3/8  α = 1/16  β = 7/16  α = 1/14  β = 3/7  
∥E∥  order  ∥E∥  order  ∥E∥  order  ∥E∥  order  
\(\frac{\pi}{20}\)  8.43e − 03  7.91e − 02  1.32e − 02  1.31e − 02  
\(\frac{\pi}{40}\)  2.12e − 03  1.99  1.98e − 02  2.00  3.69e − 03  1.84  3.67e − 03  1.84 
\(\frac{\pi}{80}\)  5.30e − 04  2.00  4.93e − 03  2.01  9.71e − 04  1.93  9.66e − 04  1.93 
\(\frac{\pi}{160}\)  1.33e − 04  1.99  1.23e − 03  2.00  2.49e − 04  1.96  2.47e − 04  1.97 
\(\frac{\pi}{320}\)  3.32e − 05  2.00  3.07e − 04  2.00  6.30e − 05  1.98  6.26e − 05  1.98 
h  PCS  PCS  PCS  

α = 1/12  β = 5/12  α = 1/10  β = 2/5  α = 1/8  β = 3/8  
∥E∥  order  ∥E∥  order  ∥E∥  order  
\(\frac{\pi}{20}\)  1.30e − 02  1.29e − 02  1.36e − 02  
\(\frac{\pi}{40}\)  3.64e − 03  1.84  3.61e − 03  1.84  3.55e − 03  1.94 
\(\frac{\pi}{80}\)  9.59e − 04  1.92  9.48e − 04  1.93  9.33e − 04  1.93 
\(\frac{\pi}{160}\)  2.46e − 04  1.96  2.43e − 04  1.96  2.39e − 04  1.96 
\(\frac{\pi}{320}\)  6.21e − 05  1.99  6.14e − 05  1.98  6.04e − 05  1.98 
Example 4.3
(Example 4.3) Maximum absolute errors and convergence orders
h  MCNS  MCNS  MCNS  MCNS  

α = 1/16  β = 7/16  α = 1/14  β = 3/7  α = 1/12  β = 5/12  α = 1/10  β = 2/5  
∥E∥  order  ∥E∥  order  ∥E∥  order  ∥E∥  order  
\(\frac{1}{20}\)  5.68e − 05  5.68e − 05  5.67e − 05  5.67e − 05  
\(\frac{1}{40}\)  1.50e − 05  1.92  1.50e − 05  1.92  1.50e − 05  1.92  1.50e − 05  1.92 
\(\frac{1}{80}\)  3.85e − 06  1.96  3.85e − 06  1.96  3.84e − 06  1.97  3.84e − 06  1.97 
\(\frac{1}{160}\)  9.75e − 07  1.98  9.74e − 06  1.98  9.73e − 07  1.98  9.72e − 07  1.98 
\(\frac{1}{320}\)  2.45e − 07  1.99  2.45e − 07  1.99  2.45e − 07  1.99  2.44e − 07  1.99 
h  MCNS  DDCS  PCS  PCS  

α = 1/8  β = 3/8  α = 1/16  β = 7/16  α = 1/14  β = 3/7  
∥E∥  order  ∥E∥  order  ∥E∥  order  ∥E∥  order  
\(\frac{1}{20}\)  5.66e − 05  2.60e − 04  1.01e − 04  1.01e − 04  
\(\frac{1}{40}\)  1.49e − 05  1.93  6.50e − 05  2.00  2.83e − 05  1.84  2.83e − 05  1.84 
\(\frac{1}{80}\)  3.83e − 06  1.96  1.63e − 05  2.00  7.48e − 06  1.92  7.48e − 06  1.93 
\(\frac{1}{160}\)  9.70e − 07  1.98  4.06e − 06  2.01  1.92e − 06  1.96  1.92e − 06  1.97 
\(\frac{1}{320}\)  2.44e − 07  1.99  1.02e − 06  1.99  4.86e − 07  1.98  4.86e − 07  1.98 
h  PCS  PCS  PCS  

α = 1/12  β = 5/12  α = 1/10  β = 2/5  α = 1/8  β = 3/8  
∥E∥  order  ∥E∥  order  ∥E∥  order  
\(\frac{1}{20}\)  1.01e − 04  1.01e − 04  1.01e − 04  
\(\frac{1}{40}\)  2.84e − 05  1.83  2.84e − 05  1.83  2.84e − 05  1.83 
\(\frac{1}{80}\)  7.49e − 06  1.92  7.49e − 06  1.92  7.50e − 06  1.92 
\(\frac{1}{160}\)  1.92e − 06  1.96  1.92e − 06  1.96  1.92e − 06  1.97 
\(\frac{1}{320}\)  4.86e − 07  1.99  4.87e − 07  1.98  4.87e − 07  1.98 
5 Conclusion
In this paper, we have developed a numerical scheme for a system of secondorder boundary value problems, which arises from secondorder obstacle problem. Our scheme is obtained by using cubic nonpolynomial spline at midknots to avoid the breakup points c and d. We have proved the unique solvability and established convergence order of our scheme. To demonstrate the numerical efficiency and to compare with other methods in the literature, three examples are presented. The numerical results illustrate that our method gives the smallest errors in all the cases.
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Authors’ contributions
All the authors contribute equally to the manuscript. All authors read and approved the final manuscript.
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None of the authors have any competing interests in the manuscript.
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