A 6-point subdivision scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems

Ghulam Mustafa; Dumitru Baleanu; Syeda Tehmina Ejaz; Kaweeta Anjum; Ali Ahmadian; Soheil Salahshour; Massimiliano Ferrara; Ghulam Mustafa; Dumitru Baleanu; Syeda Tehmina Ejaz; Kaweeta Anjum; Ali Ahmadian; Soheil Salahshour; Massimiliano Ferrara

doi:10.3934/mbe.2020346

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 6: 6659-6677. doi: 10.3934/mbe.2020346

Previous Article Next Article

Research article

A 6-point subdivision scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems

1.
Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan
2.
Department of Mathematics, Cankaya University, Ankara 06530, Turkey
3.
Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
4.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5.
Department of Mathematics, The Government Sadiq College Women University, Bahawalpur 63100, Pakistan
6.
Institute of IR 4.0, The National University of Malaysia, 43600 UKM, Bangi, Selangor, Malaysia
7.
Faculty of Engineering and Natural Sciences, Bahcesehir University, Istanbul, Turkey
8.
Department of Law, Economics and Human Sciences & Decisions Lab, University Mediterranea of Reggio Calabria, Reggio Calabria, Italy

Received: 27 June 2020 Accepted: 29 July 2020 Published: 28 September 2020

In this paper, we first present a 6-point binary interpolating subdivision scheme (BISS) which produces a C² continuous curve and 4th order of approximation. Then as an application of the scheme, we develop an iterative algorithm for the solution of 2nd order nonlinear singularly per-turbed boundary value problems (NSPBVP). The convergence of an iterative algorithm has also been presented. The 2nd order NSPBVP arising from combustion, chemical reactor theory, nuclear engi-neering, control theory, elasticity, and fluid mechanics can be solved by an iterative algorithm with 4th order of approximation.
- subdivision schemes,
- interpolation,
- approximation,
- Singularly perturbed boundary value problem,
- iterative algorithm
Citation: Ghulam Mustafa, Dumitru Baleanu, Syeda Tehmina Ejaz, Kaweeta Anjum, Ali Ahmadian, Soheil Salahshour, Massimiliano Ferrara. A 6-point subdivision scheme and its applications for the solution of 2nd order nonlinear singularly perturbed boundary value problems[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 6659-6677. doi: 10.3934/mbe.2020346

Related Papers:

Abstract

In this paper, we first present a 6-point binary interpolating subdivision scheme (BISS) which produces a C² continuous curve and 4th order of approximation. Then as an application of the scheme, we develop an iterative algorithm for the solution of 2nd order nonlinear singularly per-turbed boundary value problems (NSPBVP). The convergence of an iterative algorithm has also been presented. The 2nd order NSPBVP arising from combustion, chemical reactor theory, nuclear engi-neering, control theory, elasticity, and fluid mechanics can be solved by an iterative algorithm with 4th order of approximation.

References

[1]	G. Deslauriers, S. Dubuc, Symmetric iterative interpolation processes, Constr. Approx., 5(1989) 49-68.
[2]	A. Weissman, A 6-point interpolatory subdivision scheme for curve design, M.S. thesis, Tel-Aviv University, Tel Aviv, Israel, 1990.
[3]	B. G. Lee, Y. J. Lee, J. Yoon, Stationary binary subdivision schemes using radial basis function interpolation, Adv. Comput. Math., 25 (2006), 57-72. doi: 10.1007/s10444-004-7642-z
[4]	K. P. Ko, B. G. Lee, G. J. Yoon, A study on the mask of interpolatory symmetric subdivision schemes, Appl. Math. Comput., 187 (2007), 609-621.
[5]	J. A. Lian, On a-ary subdivision for curve design: i. 4-point and 6-point interpolatory schemes, Appl. Appl. Math., 3 (2008), 18-29.
[6]	G. Mustafa, P. Ashraf, A new 6-point ternary interpolating subdivision scheme and its differentiability, J. Inf. Comput. Sci., 5 (2010), 199-210.
[7]	G. Mustafa, J. Deng, P. Ashraf, N. A. Rehman, The mask of odd point n-ary interpolating subdivision scheme, J. Appl. Math., Article ID 205863, 2012 (2012).
[8]	G. Kanwa, A. Ghaffar, M. M. Hafeezullah, S. A. Manan, M. Rizwan, G. Rahman, Numerical solution of 2-point boundary value problem by subdivision scheme, Commun. Math. Appl., 10(1) (2019), 19-29.
[9]	G. Mustafa, S. T. Ejaz, Numerical solution of two point boundary value problems by interpolating subdivision schemes, Abstr. Appl. Anal., 2014 (2014).
[10]	R. Qu, R. P. Agarwal, Solving two point boundary value problems by interpolatory subdivision algorithms, Int J. Comput. Math., 60 (1996), 279-294. doi: 10.1080/00207169608804492
[11]	S. A., Manan, A. Ghaffar, M. Rizwan, G. Rahman, G. Kanwal, A subdivision approach to the approximate solution of 3rd order boundary value problem, Commun. Math. Appl., 9 (2018), 499-512.
[12]	S. T. Ejaz, G. Mustafa, F. Khan, Subdivision schemes based collocation algorithms for solution of fourth order boundary value problems, Math. Probl. Eng., 2015, Article ID 240138, (2015).
[13]	G. Mustafa, M. Abbas, S. T. Ejaz, A. I. Md Ismail, F. Khan, A numerical approach based on subdivision schemes for solving nonlinear fourth order boundary value problems, J. Comput. Anal. Appl., 23 (2017), 607-623.
[14]	R. Qu, R. P. Agarwal, An iterative scheme for solving nonlinear two point boundary value problems, Int. J. Comput. Math., 64 (1997), 285-302. doi: 10.1080/00207169708804591
[15]	S. T. Ejaz, G. Mustafa, A subdivision based iterative collocation algorithm for nonlinear third order boundary value problems, Adv. Math. Phys., 2016, Article ID 5026504, (2016).
[16]	G. Mustafa, S. T. Ejaz, A subdivision collocation method for solving two point boundary value problems of order three, J. Appl. Anal. Comput., 7 (2017), 942-956.
[17]	N. Dyn, Interpolatory subdivision schemes and analysis of convergence and smoothness by the formalism of laurent polynomials, in tutorials on multiresolution in geometric modeling, A. Iske, E. Quak, and M. S. Floater, Eds., pp. 51-68, Springer, Berlin, Germany, (2002).
[18]	C. Conti, K. Hormann Polynomial reproduction for univariate subdivision schemes of any arity, J. Approx. Theory, 163 (2011), 413-437. doi: 10.1016/j.jat.2010.11.002
[19]	G. Strang, Linear algebra and its applications, 4^th edition, Cengage Learning India Private Limited, ISBN-10: 81-315-0172-8, 2011.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)