Abstract

In this study, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems. The convergence analysis is discussed and the method is shown to have second order convergence. The order of convergence is improved up to fourth order using the improved end conditions. Numerical results are given to describe the efficiency of the method and compared with the method developed by Akram (2012), which shows that the present method is better.

Keywords:

Boundary layers, monotone matrices, quartic non-polynomial spline, singularly perturbed boundary value problems, uniform convergence,

References

1. Akram, G., 2012. Quartic spline solution of a third order singularly perturbed boundary value problem. Anziam J., 53: 44-58.
   CrossRef    
2. Akram, G. and S.S. Siddiqi, 2006. Solution of sixth order boundary value problems using non polynomial spline technique. Appl. Math. Comput., 181: 708-720.
   CrossRef    
3. Akram, G. and N. Mehak, 2012. Solution of a fourth order singularly perturbed boundary value problem using quintic spline. Int. Math. Forum, 7(44): 2179-2190.
   
4. Howers, F.A., 1976. Singular perturbations and differential inequalities. American Mathematical Society, Providence, Rhode Island, pp: 168.
   
5. Kelevedjiev, P., 2002. Existence of positive solutions to a singular second order boundary value problem. Nonlinear Anal-Theor., 50(8): 1107-1118.
   CrossRef    
6. Khan, A., I. Khan and T. Aziz, 2006. Sextic spline solution of singularly perturbed boundary value problem. Appl. Math. Comput., 181: 432-439.
   CrossRef    
7. Lie, J., 2008. A computational method for solving singularly perturbed two point singular boundary value problem. Int. J. Math Anal., 2: 1089-1096.
   
8. Rashidinia, J. and Z. Mahmoodi, 2007. Non-polynomial spline solution of a singularly-perturbed boundary-value problems. Int. J. Contemp. Math. Sci., 2(32): 1581-1586.
   CrossRef    
9. Roos, H.G., M. Stynes and L. Tobiska, 1996. Numerical Methods for Singularly Perturbed Difference Equation. Springer Verlag, Berlin, New York.
   CrossRef    
10. Yao, H. and M. Cui, 2007. A new algorithm for a class of singular boundary value problems. Appl. Math. Comput., 186: 1183-1191.
    CrossRef    

Competing interests

The authors have no competing interests.

Open Access Policy

This 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.

Copyright

The authors have no competing interests.