Mid-knot cubic non-polynomial spline for a system of second-order boundary value problems
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Abstract
In this paper, a mid-knot cubic non-polynomial spline is applied to obtain the numerical solution of a system of second-order boundary value problems. The numerical method is proved to be uniquely solvable and it is of second-order accuracy. We further include three examples to illustrate the accuracy of our method and to compare with other methods in the literature.
Keywords
Cubic non-polynomial spline Second-order Boundary value problem Numerical solution1 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 B-splines as basis functions to solve (1.1), while the well-known Numerov method and finite difference schemes based on the central difference have been employed in [22]. Thereafter, Al-Said 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 first-order accuracy for these methods. In [4], a two-stage method is developed where a finite difference scheme is first employed to obtain the numerical solutions at mid-knots of a uniform mesh, then a second-order interpolation is used to obtain the numerical solutions at the knots. This method is of second-order accuracy. Other proven second-order 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 mid-knots of a uniform mesh in [1, 2, 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, non-polynomial splines have also been applied to solve (1.1). Non-polynomial 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 non-polynomial splines in the literature are observed to be more accurate than that computed by polynomial splines. In fact, a cubic non-polynomial spline method has been proposed by Khan and Aziz [12] and subsequently by Siraj-ul-Islam 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 non-polynomial splines have also been used in higher-order boundary value problems, for example quartic non-polynomial spline for third-order boundary value problem [24, 26], quintic non-polynomial spline for fourth-order boundary value problem [14] and sextic non-polynomial spline for fifth-order boundary value problem [15]. Out of all these work, only [26] gives the numerical solutions of the third-order boundary value problem at mid-knots 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 non-polynomial splines, in this paper we shall develop a cubic non-polynomial spline scheme at mid-knots of a uniform mesh for the problem (1.1). The unique solvability and convergence analysis will be carried out which indicates a second-order accurate method. Finally, three examples will be presented to illustrate the numerical efficiency and the better performance over other methods in the literature.
2 Mid-knot cubic non-polynomial spline method
Throughout the paper, for any function \(v(x)\) we shall denote \(v^{(j)}(x_{i})=v^{(j)}_{i}\) and likewise \(v^{(j)}(x_{i-1/2})=v^{(j)}_{i-1/2}\). In the following, we define the cubic non-polynomial spline in terms of mid-knots 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 non-polynomial spline relation of Eq. (2.5) reduces to the well-known 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 \(t_{i}=O(h^{4}),~2\leq i\leq n-1\). If, in addition, \(\alpha=\frac{1}{12}\) (which implies \(\beta=\frac{5}{12}\)), then (2.8) yields \(t_{i}=O(h^{6}),~2\leq i\leq n-1\).
- for \(2\leq i \leq\frac{n}{4}-1\),$$ S_{i-3/2}-2S_{i-1/2}+S_{i+1/2}=h^{2} (\alpha f_{i-3/2}+2\beta f_{i-1/2}+\alpha f_{i+1/2} ); $$(2.10)
- for \(i=\frac{n}{4}\),$$\begin{aligned} &S_{n/4-3/2}-2S_{n/4-1/2}+ \bigl(1-\alpha h^{2} g_{n/4+1/2} \bigr)S_{n/4+1/2} \\ &\quad =h^{2} \bigl[\alpha f_{n/4-3/2}+2\beta f_{n/4-1/2}+ \alpha (f_{n/4+1/2}+r ) \bigr]; \end{aligned}$$(2.11)
- for \(i=\frac{n}{4}+1\),$$\begin{aligned} &S_{n/4-1/2}+ \bigl(-2-2\beta h^{2}g_{n/4+1/2} \bigr)S_{n/4+1/2}+ \bigl(1-\alpha h^{2}g_{n/4+3/2} \bigr)S_{n/4+3/2} \\ &\quad =h^{2} \bigl[\alpha f_{n/4-1/2}+2\beta (f_{n/4+1/2}+r )+\alpha (f_{n/4+3/2}+r ) \bigr]; \end{aligned}$$(2.12)
- for \(\frac{n}{4}+2\leq i \leq\frac{3n}{4}-1\),$$\begin{aligned} & \bigl(1-\alpha h^{2}g_{i-3/2} \bigr)S_{i-3/2}+ \bigl(-2-2\beta h^{2}g_{i-1/2} \bigr)S_{i-1/2}+ \bigl(1-\alpha h^{2}g_{i+1/2} \bigr)S_{i+1/2} \\ &\quad =h^{2} \bigl[\alpha (f_{i-3/2}+r )+2\beta (f_{i-1/2}+r )+\alpha (f_{i+1/2}+r ) \bigr]; \end{aligned}$$(2.13)
- for \(i=\frac{3n}{4}\),$$\begin{aligned} & \bigl(1-\alpha h^{2}g_{3n/4-3/2} \bigr)S_{3n/4-3/2}+ \bigl(-2-2\beta h^{2}g_{3n/4-1/2} \bigr)S_{3n/4-1/2}+S_{3n/4+1/2} \\ &\quad =h^{2} \bigl[\alpha (f_{3n/4-3/2}+r )+2\beta (f_{3n/4-1/2}+r )+\alpha f_{3n/4+1/2} \bigr]; \end{aligned}$$(2.14)
- for \(i=\frac{3n}{4}+1\),$$\begin{aligned} & \bigl(1-\alpha h^{2}g_{3n/4-1/2} \bigr)S_{3n/4-1/2}-2S_{3n/4+1/2}+S_{3n/4+3/2} \\ &\quad =h^{2} \bigl[\alpha (f_{3n/4-1/2}+r )+2\beta f_{3n/4+1/2}+\alpha f_{3n/4+3/2} \bigr]; \end{aligned}$$(2.15)
- for \(\frac{3n}{4}+2\leq i \leq n-1\),$$ S_{i-3/2}-2S_{i-1/2}+S_{i+1/2}=h^{2} (\alpha f_{i-3/2}+2\beta f_{i-1/2}+\alpha f_{i+1/2} ). $$(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 mid-knot cubic non-polynomial 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 fourth-order convergence method. However, the solution of problem (1.1) exists continuously only up to the second derivative. Therefore, the numerical method is only second-order 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, 5, 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 well-known 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 second-order boundary value problems.
Example 4.1
([19])
(Example 4.1) Maximum absolute errors
Methods | h = π/20 | h = π/40 | h = π/80 |
---|---|---|---|
Mid-knot cubic non-polynomial spline α = 1/8, β = 3/8 | 2.40e − 04 | 6.34e − 05 | 1.63e − 05 |
Mid-knot cubic non-polynomial spline α = 1/10, β = 2/5 | 3.44e − 04 | 9.11e − 05 | 2.34e − 05 |
Mid-knot cubic non-polynomial spline α = 1/12, β = 5/12 | 4.14e − 04 | 1.09e − 04 | 2.81e − 05 |
Mid-knot cubic non-polynomial spline α = 1/14, β = 3/7 | 4.64e − 04 | 1.23e − 04 | 3.15e − 05 |
Mid-knot cubic non-polynomial 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 |
Collocation-cubic 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, 2, 3, 4, 5, 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 second-order boundary value problems, which arises from second-order obstacle problem. Our scheme is obtained by using cubic non-polynomial spline at mid-knots 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.
Notes
Acknowledgements
Not applicable.
Availability of data and materials
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Authors’ contributions
All the authors contribute equally to the manuscript. All authors read and approved the final manuscript.
Funding
Not applicable.
Competing interests
None of the authors have any competing interests in the manuscript.
References
- 1.Al-Said, E.A.: Spline solutions for system of second-order boundary-value problems. Int. J. Comput. Math. 62, 143–154 (1996) MathSciNetCrossRefGoogle Scholar
- 2.Al-Said, E.A.: Spline methods for solving system of second-order boundary-value problems. Int. J. Comput. Math. 70, 717–727 (1999) MathSciNetCrossRefGoogle Scholar
- 3.Al-Said, E.A.: The use of cubic splines in the numerical solution of a system of second-order boundary value problems. Comput. Math. Appl. 42, 861–869 (2001) MathSciNetCrossRefGoogle Scholar
- 4.Al-Said, E.A., Noor, M.A.: Modified Numerov method for solving system of second-order boundary-value problems. Korean J. Comput. Appl. Math. 8, 129–136 (2001) MathSciNetzbMATHGoogle Scholar
- 5.Al-Said, E.A., Noor, M.A., Al-Shejari, A.A.: Numerical solutions for system of second order boundary value problems. Korean J. Comput. Appl. Math. 5, 659–667 (1998) MathSciNetzbMATHGoogle Scholar
- 6.Aziz, T., Khan, A., Khan, I.: Quintic splines method for second-order boundary value problems. Int. J. Comput. Math. 85, 735–743 (2008) MathSciNetCrossRefGoogle Scholar
- 7.Baiocchi, C., Capelo, A.: Variational and Quasi-Variational Inequalities. Wiley, New York (1984) zbMATHGoogle Scholar
- 8.Chen, F., Wong, P.J.Y.: Deficient discrete cubic spline solution for a system of second order boundary value problems. Numer. Algorithms 66, 793–809 (2014) MathSciNetCrossRefGoogle Scholar
- 9.Cottle, R.W., Giannessi, F., Lions, J.L.: Variational Inequalities and Complementarity Problems: Theory and Applications. Wiley, New York (1980) Google Scholar
- 10.Crank, J.: Free and Moving Boundary Problems. Clarendon Press, Oxford (1984) zbMATHGoogle Scholar
- 11.Fröberg, C.: Numerical Mathematics, Theory and Computer Applications. Benjamin/Commings, Reading (1985) zbMATHGoogle Scholar
- 12.Khan, A., Aziz, T.: Parametric cubic spline approach to the solution of a system of second-order boundary-value problems. J. Optim. Theory Appl. 118, 45–54 (2003) MathSciNetCrossRefGoogle Scholar
- 13.Khan, A., Khan, I., Aziz, T.: A survey on parametric spline function approximation. Appl. Math. Comput. 171, 983–1003 (2005) MathSciNetzbMATHGoogle Scholar
- 14.Khan, A., Noor, M.A., Aziz, T.: Parametric quintic-spline approach to the solution of a system of fourth-order boundary-value problems. J. Optim. Theory Appl. 122, 309–322 (2004) MathSciNetCrossRefGoogle Scholar
- 15.Khan Siraj-ul-Islam, M.A., Tirmizi, I.A., Twizell, E.H., Ashraf, S.: A class of methods based on non-polynomial sextic spline functions for the solution of a special fifth-order boundary-value problems. J. Math. Anal. Appl. 321, 651–660 (2006) MathSciNetCrossRefGoogle Scholar
- 16.Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity. SIAM, Philadelphia (1988) CrossRefGoogle Scholar
- 17.Lewy, H., Stampacchia, G.: On the regularity of the solution of a variational inequality. Commun. Pure Appl. Math. 22, 153–188 (1960) MathSciNetCrossRefGoogle Scholar
- 18.Momani, S.: Solving a system of second order obstacle problems by a modified decomposition method. Appl. Math. E-Notes 6, 141–148 (2006) MathSciNetzbMATHGoogle Scholar
- 19.Noor, M.A., Khalifa, A.K.: Cubic splines collocation methods for unilateral problems. Int. J. Eng. Sci. 25, 1525–1530 (1987) MathSciNetCrossRefGoogle Scholar
- 20.Noor, M.A., Noor, K.I., Rafiq, M., Al-Said, E.A.: Variational iteration method for solving a system of second-order boundary value problems. Int. J. Nonlinear Sci. Numer. Simul. 11, 1109–1120 (2010) CrossRefGoogle Scholar
- 21.Noor, M.A., Noor, K.I., Rassias, Th.: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993) MathSciNetCrossRefGoogle Scholar
- 22.Noor, M.A., Tirmizi, S.I.A.: Finite difference technique for solving obstacle problems. Appl. Math. Lett. 1, 267–271 (1988) MathSciNetCrossRefGoogle Scholar
- 23.Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987) zbMATHGoogle Scholar
- 24.Siraj-ul-Islam, Khan, M.A., Tirmizi, I.A., Twizell, E.H.: Non polynomial spline approach to the solution of a system of third-order boundary-value problems. Appl. Math. Comput. 168, 152–163 (2005) MathSciNetzbMATHGoogle Scholar
- 25.Siraj-ul-Islam, Tirmizi, I.A.: Nonpolynomial spline approach to the solution of a system of second-order boundary-value problems. Appl. Math. Comput. 173, 1208–1218 (2006) MathSciNetzbMATHGoogle Scholar
- 26.Siraj-ul-Islam, Tirmizi, I.A., Khan, M.A.: Quartic non-polynomial spline approach to the solution of a system of third-order boundary-value problems. J. Math. Anal. Appl. 335, 1095–1104 (2007) MathSciNetCrossRefGoogle Scholar
- 27.Usmani, R.A.: Bounds for the solution of a second order differential equation with mixed boundary conditions. J. Eng. Math. 9, 159–164 (1975) MathSciNetCrossRefGoogle Scholar
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