A New Family of High-order Difference Schemes for the Solution of Second Order Boundary Value Problems

Document Type : Research Paper


1 Department of Applied Mathematics, Faculty of Mathematical Science, University of Kashan, Kashan, Iran

2 Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran


Many problems in chemistry, nanotechnology, biology, natural science, chemical physics and engineering are modeled by two point boundary value problems. In general, analytical solution of these problems does not exist. In this paper, we propose a new class of high-order accurate methods for solving special second order nonlinear two point boundary value problems. Local truncation errors of these methods are discussed. To illustrate the potential of the new methods, we apply them for solving some well-known problems, including Troesch’s problem, Bratu’s problem and certain singularly perturbed problem. Bratu’s problem and Troech’s problems, may be used to model some chemical reaction-diffusion and heat transfer processes. We also compare the results of this work with some existing results in the literature and show that the new methods are efficient and applicable.


Main Subjects

1. H. B. Keller, Numerical Methods for Two−Points Boundary−Value Problems,
Dover, New York, 1992.
2. Z. A. Majid, N. A. Azmi, M. Suleiman, Solving second order ordinary
differential equation ns using two point four step direct implicit block method,
Eur. J. Sci. Res. 31 (2009) 29−36.
3. A. Saadatmandi, M. R. Azizi, Chebyshev finite difference method for a twopoint
boundary value problems with applications to chemical reactor theory,
Iranian J. Math. Chem. 3 (2012) 1−7.
4. M. Dehghan, A. Saadatmandi, The numerical solution of a nonlinear system of
second-order boundary value problems using the sinc-collocation method, Math.
Comput. Modelling 46 (2007) 1434−1441.
5. S Yeganeh, Y. Ordokhani, A. Saadatmandi, A sinc-collocation method for
second-order boundary value problems of nonlinear integro-differential
equation, J. Inform. Comput. Sci. 7 (2012) 151−160.
6. F. W. Gelu, G. F. Duressa, T. A. Bullo, Sixth-order compact finite difference
method for singularly perturbed 1D reaction diffusion problems, J. Taibah Univ.
Sci. 11 (2017) 302−308.
7. U. Erdugan, T. Ozis, A smart nonstandard finite difference scheme for second
order nonlinear boundary value probles, J. Comput. Phys. 230 (2011)
8. R. E. Mickens, Advances in the Applications of Nonstandard Finite Difference
Schemes, Wiley−Interscience, Singapore, 2005.
9. P. K. Pandey, Rational finite difference approximation of high order accuracy
for nonlinear two point boundary value problems, Sains Malays 43 (2014)
10. P. K. Pandey, A non-classical finite difference method for solving two point
boundary value problems, Pac. J. Sci. Technol. 14 (2013) 147−152.
11. H. Ramos, A non-standard explicit integration scheme for initial-value
problems, Appl. Math. Comput. 189 (2007) 710−718.
12. E. S. Weibel, On the confinement of a plasma by magnetostatic fields, Phys.
Fluids 2 (1959) 52−56.
13. D. Gidaspow, B. S. Baker, A model for discharge of storage batteries, J.
Electrochem. Soc. 120 (1973) 1005−1010.
14. A. Kouibia, M. Pasadas, Z. Belhaj, A. Hananel, The variational spline method
for solving Troesch’s problem, J. Math. Chem. 53 (2015) 868−879.
15. S. M. Roberts, J. S. Shipman, On the closed form solution of Troesch’s problem,
J. Comput. Phys. 21 (1976) 291−304.
16. J. P. Chiou, T. Y. Na, On the solution of Troesch’s nonlinear two-point
boundary value problem using an initial value method, J. Comput. Phys. 19
(1975) 311−316.
17. H. Temimi, A discontinuous Galerkin finite element method for solving the
Troesch’s problem, Appl. Math. Comput. 219 (2012) 521–529.
18. S. H. Chang, A variational iteration method for solving Troesch’s problem, J.
Comput. Appl. Math. 234 (2010) 3043−3047.
19. S. H. Chang, Numerical solution of Troesch’s problem by simple shooting
method, Appl. Math. Comput. 216 (2010) 3303−3306.
20. S. A. Khuri, A. Sayfy, Troesch’s problem: A B-spline collocation approach,
Math. Comput. Modelling 54 (2011) 1907−1918.
21. A. Saadatmandi, T. Abdolahi-Niasar, Numerical solution of Troesch's problem
using Christov rational functions, Comput. Methods Differ. Equ. 3 (2015)
22. M. Zarebnia, M. Sajjadian, The sinc-Galerkin method for solving Troesch’s
problem, Math. Comput. Modelling 56 (2012) 218−228.
23. H. Temimi, M. Ben-Romdhane, A. R. Ansari, G. I. Shishkin, Finite difference
numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh,
Calcolo 54 (2017) 225−242.
24. H. N. Hassan, M. A. El-Tawilb, An efficient analytic approach for solving twopoint
nonlinear boundary value problems by homotopy analysis method, Math.
Methods Appl. Sci. 34 (2011) 977−989.
25. A. M. Wazwaz, A reliable study for extensions of the Bratu problem with
boundary conditions, Math. Methods Appl. Sci. 35 (2012) 845−856.
26. J. Rashidinia, K. Maleknejad, N. Taheri, Sinc-Galerkin method for numerical
solution of the Bratu’s problems, Numer. Algorithms 62 (2013) 1−11.
27. H. Caglar, N. Caglar, M. Ozer, A. Valarıstos, A. N. Anagnostopoulos, B-spline
method for solving Bratu’s problem, Int. J. Comput. Math. 87 (2010)
28. E. Deeba, S. A. Khuri, S. Xie, An algorithm for solving boundary value
problems, J. Comput. Phys. 159 (2000) 125−138.
29. J. Karkowski, Numerical experiments with the Bratu equation in one, two and
three dimensions, Comput. Appl. Math. 32 (2013) 231−244.
30. H. Temimi, M. Ben-Romdhane, An iterative finite difference method for solving
Bratu’s problem, J. Comput. Appl. Math. 292 (2016) 76−82.
31. J. Rashidinia, R. Mohammadi, S. H. Moatamedoshariati, Quintic spline methods
for the solution of Singularly perturbed boundary-value problems, Int. J.
Comput. Methods Eng. Sci. Mech. 11 (2010) 247−257.
32. A. Khan, I. Khan, T. Aziz, Sextic spline solution of a singularly perturbed
boundary value problems, Appl. Math. Comput. 181 (2006) 432−439.
33. A. Saadatmandi, Z. Akbari, Transformed Hermite functions on a finite interval
and their applications to a class of singular boundary value problems, Comput.
Appl. Math. 36 (2017) 1085−1098.
34. A. Saadatmandi, N. Nafar, S. P. Toufighi, Numerical study on the reaction cum
diffusion process in a spherical biocatalyst, Iranian J. Math. Chem. 5 (2014)
35. T. Caraballo, M. Herrera-Cobos, P. Marín-Rubio, An iterative method for nonautonomous
nonlocal reaction-diffusion equations, Appl. Math. Nonlinear Sci. 2
(2017) 73−82.
36. F. Balibrea, On problems of Topological Dynamics in non-autonomous discrete
systems, Appl. Math. Nonlinear Sci. 1 (2016) 391−404.
37. E Babolian, A Eftekhari, A Saadatmandi, A sinc-Galerkin approximate solution
of the reaction–diffusion process in an immobilized biocatalyst pellet, MATCH
Commun. Math. Comput. Chem. 71 (2014) 681−697.