- Research
- Open Access
Solving third-order boundary value problems with quartic splines
- P. K. Pandey^{1}Email author
- Received: 14 August 2015
- Accepted: 3 March 2016
- Published: 15 March 2016
Abstract
In this article, we present a novel second order numerical method for solving third order boundary value problems using the quartic polynomial splines. We establish the convergence of the method. We present numerical experiments to demonstrate the efficiency of the method and validity of our second order method, which shows that present method gives better results.
Keywords
- Boundary-value problems
- Finite-difference methods
- Obstacle problems
- Quartic polynomial splines
Mathematics Subject Classification
- 65L10
- 65L12
- 65L20
Background
In environments and in most other areas of natural and applied sciences, the differential equations that govern the behavior of model systems are well-known. For instance, to describe the evolution of physical phenomena in fluctuating environments governed by third order differential equation (Ahmad et al. 2012). The study of aero elasticity, sandwich beam analysis and beam deflection theory, electromagnetic waves, theory of thin film flow and incompressible flows and regularization of the Cauchy problem for one-dimensional hyperbolic conservation laws (Bressan 2000) are some other model systems in natural and applied sciences where the third order boundary value problems arise.
The theoretical concepts of existence, uniqueness and convergence of the solution and some specific solution of problem (1) can be found in the literature (Howes 1982; Agarwal 1986; Gupta and Lakshmikantham 1991; Gregus 1987; Murty and Rao 1992; Feckan 1994; Henderson and Prasad 2001; Li 2010). The specific assumption to further ensure existence and uniqueness of the solution to problem (1) will not be considered. Thus the existence and uniqueness of the solution to problem (1) is assumed. Further we assume that problem (1) is well pose. The emphasis in this article will be on the development of an efficient numerical method to deal with approximate numerical solution of the third order boundary value problem.
The quality of a numerical method depends on the accuracy of the method to a great extent. Some efficient and accurate numerical methods for solving higher order boundary value problems are available in literature. Some researchers have studied and solved in particular third order boundary value problems with different boundary conditions using different methods for instance some literary work in Finite Difference Method (Al-Said 2001), Quintic Splines (Khan and Aziz 2003), Non polynomial spline method (Islam et al. 2005, 2007; Srivastava and Kumar 2012), Quartic B-splines (Gao and Chi 2006), Haar wavelets method (Fazal-i-Haq and Ali 2011), Collocation quantic spline (Noor and Khalifa 1994), Reproducing Kernel Method (Li and Wu 2012) and references therein can be found. With advent of computers it gained important to develop more accurate numerical methods to solve higher order boundary value problems. Hence, the purpose of this article is to develop an efficient numerical method for solution of third order boundary value problems (1).
We present our work in this article as follows. In the next section we derive a finite difference method. In section “Convergence analysis”, we discuss convergence of the proposed method under appropriate condition. The application of the proposed method on the test problems and illustrative numerical results so produced to show the efficiency in section “Numerical results”. Discussion and conclusion on the performance of the proposed method present in section “Conclusion”.
The difference method
Convergence analysis
Thus from Eq. (22) it follows that \(\Vert {\mathbf{e}} \Vert \rightarrow 0\) as \(h \rightarrow 0\). This establishes the convergence of the method (7) and the order of convergence of method (7) is at least \(O(h^2)\).
Numerical results
Maximum absolute error (Problem 1)
Maximum absolute error (Problem 2)
Maximum absolute error (Problem 3)
N | Maximum absolute error | |||
---|---|---|---|---|
MAEI | MAEM | MAEE | MAE | |
8 | .74443180 (−3) | .24691788 (−1) | .56693217 (−1) | .56693217 (−1) |
16 | .11904127 (−4) | .11839149 (−3) | .42345375 (−4) | .11839149 (−3) |
32 | .29750796 (−4) | .28597564 (−4) | .19483268 (−4) | .29750796 (−4) |
64 | .74037612 (−5) | .58218491 (−4) | .74401498 (−4) | .74401498 (−4) |
Maximum absolute error (Problem 3)
Method | Maximum absolute error | ||
---|---|---|---|
N = 16 | N = 32 | N = 64 | |
Equation (7) | .119 (−3) | .297 (−4) | .744 (−4) |
Al-Said (2001) | .196 (−3) | .489 (−4) | .122 (−4) |
Islam et al. (2005) | .712 (−3) | .405 (−3) | .222 (−3) |
Gao and Chi (2006) | .113 (−2) | .530 (−3) | .252 (−3) |
Noor and Al-Said (2004) | .115 (−2) | .532 (−3) | .256 (−3) |
Li and Wu (2012) | .118 (−2) | .547 (−3) | .262 (−3) |
Al-Said and Noor (2003) | .123 (−2) | .553 (−3) | .261 (−3) |
Noor and Khalifa (1994) | .126 (−2) | .560 (−3) | .310 (−3) |
Al-Said et al. (1996) | .689 (−2) | .711 (−2) | .727 (−3) |
We have described a numerical method for numerical solution of third order boundary value problem and three model problems including an obstacle problem considered to test the performance of the proposed method. Numerical result for examples for different values of N which is presented in tables, the maximum absolute errors in solution decreases with decrease in step size h. Also from the numerical results in Table 4, is clear that the new method (7) outperforms the existing methods. On the other hand, it is evident that method (7) is convergent and the rate of convergence is at least quadratic.
Conclusion
A finite difference method to find the numerical solution of third order boundary value problems has been developed. At nodal point \(x = x_{i-\frac{1}{2}}, i = 1, 2,\ldots ,N\) we have obtained a system of algebraic equations given by (7). Thus we have a system of linear equations if source function f(x, u) is linear otherwise system of nonlinear equations. The propose method produces good approximate numerical value of the solution for model problems and it is computationally efficient and accurate method. The idea presented in this article leads to the possibility to develop finite difference methods for the numerical solution of higher odd order boundary value problems. Works in these directions are in progress.
Declarations
Acknowledgements
It is not possible for us to pay processing charges of journal. I greatly acknowledge the support from Springer Plus in term of waivers of processing fee.
Competing interests
All 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
References
- Agarwal RP (1986) Boundary value problems for higher order differential equations. World Scientific, SingaporeView ArticleGoogle Scholar
- Ahmad B, Nieto JJ, Alsaedi A, El-Shahed M (2012) A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal Real World Appl 13:599–606View ArticleGoogle Scholar
- Al-Said EA, Noor MA, Khalifa AK (1996) Finite difference scheme for variational inequalities. J Optim Theory Appl 89(2):453–459View ArticleGoogle Scholar
- Al-Said EA (2001) Numerical solutions for system of third-order boundary value problems. Int J Comput Math 78(1):111–121View ArticleGoogle Scholar
- Al-Said EA, Noor MA (2003) Cubic splines method for a system of third-order boundary value problems. Appl Math Comput 142(2–3):195–204View ArticleGoogle Scholar
- Bressan A (2000) Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. Oxford University Press, LondonGoogle Scholar
- Fazal-i-Haq IH, Ali A (2011) A Haar wavelets based numerical method for third-order boundary and initial value problems. World Appl Sci J 13(10):2244–2251Google Scholar
- Feckan M (1994) Singularly perturbed higher order boundary value problems. J Differ Equ 111(1):79–102View ArticleGoogle Scholar
- Gao F, Chi CM (2006) Solving third-order obstacle problems with quartic B-splines. Appl Math Comput 180(1):270–274View ArticleGoogle Scholar
- Gregus M (1987) Third order linear differential equations. Series: mathematics and its applications, vol 22. Springer, NetherlandsView ArticleGoogle Scholar
- Gupta CP, Lakshmikantham V (1991) Existence and uniqueness theorems for a third-order three point boundary value problem. Nonlinear Anal Theory Methods Appl 16(11):949–957View ArticleGoogle Scholar
- Henderson J, Prasad KR (2001) Existence and uniqueness of solutions of three-point boundary value problems on time scales. Nonlinear Stud 8:1–12Google Scholar
- Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, New York 10011Google Scholar
- Howes FA (1982) Differential inequalities of higher order and the asymptotic solution of the nonlinear boundary value problems. SIAM J Math Anal 13(1):61–80View ArticleGoogle Scholar
- Islam S, Khan MA, Tirmizi IA, Twizell EH (2005) Non-polynomial splines approach to the solution of a system of third-order boundary-value problems. Appl Math Comput 168(1):152–163View ArticleGoogle Scholar
- Islam SU, Tirmizi IA, Khan MA (2007) Quartic non-polynomial spline approach to the solution of a system of third-order boundary-value problems. J Math Anal Appl 335(2):1095–1104View ArticleGoogle Scholar
- Jain MK, Iyenger SRK, Jain RK (1987) Numerical methods for scientific and engineering computation, 2nd edn. Willey Eastern Limited, New DelhiGoogle Scholar
- Khan A, Aziz T (2003) The numerical solution of third-order boundary-value problems using quintic splines. Appl Math Comput 137(2–3):253–260View ArticleGoogle Scholar
- Li YX (2010) Positive periodic solutions for fully third-order ordinary differential equations. Comput Math Appl 59:3464–3471View ArticleGoogle Scholar
- Li X, Wu B (2012) Reproducing kernel method for singular multi-point boundary value problems. Math Sci 6:16. doi:10.1186/2251-7456-6-16 View ArticleGoogle Scholar
- Murty KN, Rao YS (1992) A theory for existence and uniqueness of solutions to three-point boundary value problems. J Math Anal Appl 167(1):43–48View ArticleGoogle Scholar
- Noor MA, Al-Said EA (2004) Quartic splines solutions of third-order obstacle problems. Appl Math Comput 153(2):307–316View ArticleGoogle Scholar
- Noor MA, Khalifa AK (1994) A numerical approach for odd-order obstacle problems. Int J Comput Math 54(1):109–116View ArticleGoogle Scholar
- Pandey PK. An efficient numerical method for the solution of third order boundary value problem in ordinary differential equations (to appear)Google Scholar
- Srivastava PK, Kumar M (2012) Numerical algorithm based on quintic nonpolynomial spline for solving third-order boundary value problems associated with draining and coating flow. Chin Ann Math Ser B 33(6):831–840View ArticleGoogle Scholar
- Usmani RA, Sakai M (1984) Quartic spline solutions for two-point boundary problems involving third order differential equations. J Math Phys Sci 18:365–380Google Scholar
- Varga RS (2000) Matrix iterative analysis, second revised and expanded edition. Springer, HeidelbergView ArticleGoogle Scholar