IJPAM: Volume 110, No. 2 (2016)




Ahmed Salem Heilat$^1$, Ahmad Izani Md. Ismail$^2$
$^{1,2}$School of Mathematical Sciences
Universiti Sains Malaysia
11800, Penang, MALAYSIA


A method based on hybrid cubic B-spline method (HCBSM) is developed, analyzed and applied to solve second-order non-linear two-point boundary value problems. In this method, a free parameter, $\gamma$, plays an important role in producing accurate results. Tests on four examples and a comparison of the results with the results obtained using cubic B-spline, extended cubic B-spline and shooting methods indicated that HCBSM is a feasible and accurate method.


Received: June 26, 2016
Revised: October 15, 2016
Published: November 5, 2016

AMS Classification, Key Words

AMS Subject Classification: Key Words and Phrases: two-point boundary value problems, cubic b-spline, trigonometric cubic b-spline, hybrid cubic b-spline, quasilinearization

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Ha, Sung. N., A nonlinear shooting method for two-point boundary value problems, Computers and Mathematics with Applications, 42, 10(2001), 1411-1420, doi: https://doi.org/10.1016/S0898-1221(01)00250-4.

Burden, R. L., JD Faires Numerical Analysis, Brooks-Cole.Pub, Boston:United States (1985).‏

Mo, Lu-Feng, and Shu-Qiang Wang, A variational approach to nonlinear two-point boundary value problems, Nonlinear Analysis: Theory, Methods and Applications 71, 12(2009), e834-e838, doi: https://doi.org/10.1016/j.na.2008.12.006.

Geng, Fazhan, and Minggen Cui., A novel method for nonlinear two-point boundary value problems: combination of ADM and RKM, Applied Mathematics and Computation 217, 9(2011), 4676-4681, doi: https://doi.org/10.1016/j.amc.2010.11.020.

Chun, Changbum, and Rathinasamy Sakthivel, Homotopy perturbation technique for solving two-point boundary value problems–comparison with other methods, Computer Physics Communications, 181, 6(2010), 1021-1024, doi: https://doi.org/10.1016/j.cpc.2010.02.007.

Abbasbandy, Saeid, Babak Azarnavid, and Mohammed S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 279, (2015), 293-305, doi: https://doi.org/10.1016/j.cam.2014.11.014.

Zhang, Ruimin, and Yingzhen Lin, A novel method for nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 282, (2015), 77-82, doi: https://doi.org/10.1016/j.cam.2014.12.035.

Lin, Yingzhen, Jing Niu, and Minggen Cui, A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space, Applied Mathematics and Computation, 218, 14(2012), 7362-7368, doi: https://doi.org/10.1016/j.amc.2011.11.009.

Aziz, Imran, and Božidar Šarler, The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets, Mathematical and Computer Modelling, 52, 9(2010), 1577-1590, doi: https://doi.org/10.1016/j.mcm.2010.06.023.

Cuomo, Salvatore, and A. Marasco, A numerical approach to nonlinear two-point boundary value problems for ODEs, Computers and Mathematics with Applications, 55, 11(2008), 2476-2489, doi: https://doi.org/10.1016/j.camwa.2007.10.002.

Jain, M. K., Sachin Sharma, and R. K. Mohanty, High accuracy variable mesh method for nonlinear two-point boundary value problems in divergence form, Applied Mathematics and Computation, 273, (2016), 885-896, doi: https://doi.org/10.1016/j.amc.2015.10.030.

El-Kalla, I. L., Piece-wise continuous solution to a class of nonlinear boundary value problems, Ain Shams Engineering Journal 4, 2(2013), 325-331, doi: https://doi.org/10.1016/j.asej.2012.08.011.

Ahsan, Muhammad, and Sarah Farrukh, A new type of shooting method for nonlinear boundary value problems, Alexandria Engineering Journal 52, 4(2013), 801-805, doi: https://doi.org/10.1016/j.aej.2013.07.001.

Kanth, ASV Ravi, Cubic spline polynomial for non-linear singular two-point boundary value problems, Applied mathematics and computation 189, 2(2007), 2017-2022, doi: https://doi.org/10.1016/j.amc.2007.01.002.

Caglar, Hikmet, Nazan Caglar, and Khaled Elfaituri, B-spline interpolation compared with finite difference, finite element and finite volume methods which applied to two-point boundary value problems, Applied Mathematics and computation 175, 1(2006), 72-79, doi: https://doi.org/10.1016/j.amc.2005.07.019.

Hamid, N. N., Ahmad Abd Majid, and Ahmad Izani Md Ismail, Cubic trigonometric B-spline applied to linear two-point boundary value problems of order two, World Academic of Science, Engineering and Technology 47, (2010), 478-803, https://waset.org/publications/13902.

Hamid, Nur Nadiah Abd, Ahmad Abd, and Ahmad Izani Md Ismail, Extended Cubic B-spline Interpolation Method Applied to Linear Two-Point Boundary Value Problems, World Academy of Science, Engineering and Technology 62, (2010), https://waset.org/publications/12332.

Joan Goh, B-splines for initial and boundary value problems, PHD thesis, Universiti Sains Malaysia (2012), elib.usm.my/cgi-bin/koha/opac-detail.pl?biblionumber=635095.

Khuri, S. A., and Ali Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology, Mathematical and Computer Modelling 52, 3(2010), 626-636, doi: https://doi.org/10.1016/j.mcm.2010.04.009.

Mat Zin, Shazalina, et al., Application of Hybrid Cubic B-Spline Collocation Approach for Solving a Generalized Nonlinear Klien-Gordon Equation, Mathematical Problems in Engineering 2014 (2014), doi: https://doi.org/10.1155/2014/108560.

Rao, S. C. S., and M. Kumar, B-spline collocation method for nonlinear singularly-perturbed two-point boundary-value problems, Journal of optimization theory and applications, 134, 1(2007), 91-105, doi: https://doi.org/s10957-007-9200-6.

Pandey, Pramod, Solving nonlinear two point boundary value problems using exponential finite difference method, Boletim da Sociedade Paranaense de Matemática 34, 1(2014), 33-44, doi: https://doi.org/10.5269/bspm.v34i1.23036.

How to Cite?

DOI: 10.12732/ijpam.v110i2.11 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 110
Issue: 2
Pages: 369 - 381

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