An Advantageous Numerical Method for Solution of Linear Differential Equations by Stancu Polynomials
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Abstract
In this study, a numerical method that is alternative to the Bernstein collocation method has been investigated for solution of the linear differential equations. The theory of the method has been constituted by considering the Stancu polynomials and their algebric properties. The applicability of the method has been indicated on initial and boundary value problems. In addition, the numerical results of the proposed method have been compared with the numerical results of the known method had the best approximation in the past studies. Therefore, whether usability and efficiency of the proposed method is or not has been presented.
2020 Mathematics subject classification: 41A10, 65L05, 65L10 , 65L60
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Stancu DD. Approximation of functions by a new class of linear polynomial operators. Rev Roumaine Math Pure Appl. 1968;13:1173–94. Available from: https://cir.nii.ac.jp/crid/1370565164571100049
Altomare P, Campiti M. Korovkin-type approximation theory and its applications. Berlin: Walter de Gruyter; 1994. Available from: https://ricerca.uniba.it/handle/11586/67184
Stancu DD. Approximation of functions by means of a new generalized Bernstein operator. Calcolo. 1983;20:211–229. Available from: https://link.springer.com/article/10.1007/BF02575593
Bernstein S. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun Soc Math Kharkow. 1912;13:1–2. Available from: https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=khmo&paperid=107&option_lang=rus
Mishra VN, Gandhi RB. Study of sensitivity of parameters of Bernstein–Stancu operators. Iran J Sci Technol Trans Sci. 2019;43:2891–7. Available from: https://link.springer.com/article/10.1007/s40995-019-00761-x
Yüzbaşı Ş, Güler H. Bernstein collocation method for solving the first-order nonlinear differential equations with the mixed non-linear conditions. Math Comput Appl. 2015;20:160–73. Available from: http://dx.doi.org/10.19029/mca-2015-014
İşler Acar N, Daşcıoğlu A. A projection method for linear Fredholm-Volterra integro-differential equations. J Taibah Univ Sci. 2019;13:644–50. Available from: https://doi.org/10.1080/16583655.2019.1616962
Jafari H, Tajadodi H, Ganji RM. A numerical approach for solving variable order differential equations based on Bernstein polynomials. Comput Math Methods. 2019;1:1–11. Available from: https://doi.org/10.1002/cmm4.1055
Ishtiaq A. Bernstein collocation method for neutral type functional differential equations. Math Biosci Eng. 2021;18:2764–74. Available from: https://www.aimspress.com/aimspress-data/mbe/2021/3/PDF/mbe-18-03-140.pdf
Olagunju AS, Joseph FL, Atanyi YE. Performance comparison of spread and Bernstein basis in the solution of fractional differential equations via collocation method. Trans Niger Assoc Math Phys. 2022;18:125–32. https://nampjournals.org.ng/index.php/tnamp/article/view/163
Shahni J, Singh R. Numerical results of Emden-Fowler boundary value problems with derivative dependence using the Bernstein collocation method. Eng Comput. 2022;38:371–80. Available from: https://link.springer.com/article/10.1007/s00366-020-01155-z
Shaqul MI, Hossain MB. Numerical solutions of eighth order BVP by the Galerkin residual technique with Bernstein and Legendre polynomials. Appl Math Comput. 2015;261:48–59. Available from: https://doi.org/10.1016/j.amc.2015.03.091
Yüzbaşı Ş, Karaay M. A Galerkin-like method for solving linear functional differential equations under initial conditions. Turk J Math. 2020;44:85–97. Available from: https://journals.tubitak.gov.tr/math/vol44/iss1/5/
Sohel MN, Islam MS, Islam MS. Galerkin residual correction for fourth-order BVP. J Appl Math Comput. 2022;6:127–38. Available from: https://www.hillpublisher.com/UpFile/202203/20220325174704.pdf
Doha EH, Bhrawy AH, Saker MA. On the derivatives of Bernstein polynomials: An application for the solution of high even-order differential equations. Bound Value Probl. 2011;2011:1–16. Available from: https://link.springer.com/content/pdf/10.1155/2011/829543.pdf
Khalil H, Khan RA, Rashidi MM. Bernstein polynomials and applications to fractional differential equations. Comput Methods Differ Equ. 2015;3:14–35. Available from: https://cmde.tabrizu.ac.ir/article_3798_464.html
Parand K, Hossayni SA, Rad JA. Operation matrix method based on Bernstein polynomials for the Riccati differential equation and Volterra population model. Appl Math Model. 2016;40:993–1011. Available from: https://doi.org/10.1016/j.apm.2015.07.002
Pirabaharan P, Chandrakumar RD. A computational method for solving a class of singular boundary value problems arising in science and engineering. Egypt J Basic Appl Sci. 2016;3:383–91. Available from: https://doi.org/10.1016/j.ejbas.2016.09.004
Tabrizidooz HR, Shabanpanah K. Bernstein polynomial basis for numerical solution of boundary value problems. Numer Algor. 2018;77:211–28. Available from: https://link.springer.com/article/10.1007/s11075-017-0311-3
Khataybeh SN, Hashim I, Alshbool M. Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations. J King Saud Univ Sci. 2019;31:822–6. Available from: https://doi.org/10.1016/j.jksus.2018.05.002
Kadkhoda N. A numerical approach for solving variable order differential equations using Bernstein polynomials. Alex Eng J. 2020;59:3041–7. Available from: https://doi.org/10.1016/j.aej.2020.05.009
Quasim AF, Al-Rawi ES. Adomian decomposition method with modified Bernstein polynomials for solving ordinary and partial differential equations. J Appl Math. 2018;2018:1–9. Available from: https://doi.org/10.1155/2018/1803107
Yousif AN, Qasim AF. A novel iterative method based on Bernstein-Adomian polynomials to solve non-linear differential equations. Open Access Library J. 2020;7:1–12. Available from: http://www.scirp.org/journal/PaperInformation.aspx?PaperID=99890abstract
Farouki RT, Rajan VT. Algorithms for polynomials in Bernstein form. Comput Aided Geom Design. 1988;5:1–26. Available from: https://doi.org/10.1016/0167-8396(88)90016-7
Akyüz-Daşcıoğlu A, İşler Acar N. Bernstein collocation method for solving linear differential equations. GU J Sci. 2013;26:527–34. Available from: https://dergipark.org.tr/en/download/article-file/83614
İşler Acar N. Diferansiyel, integral ve integro-diferansiyel denklemler için Bernstein yaklaşımı. Denizli: Pamukkale Üniversitesi, Fen Bilimleri Enstitüsü; 2015. Available from: https://hdl.handle.net/11499/1721
Mestrovic M. The modified decomposition method for eighth-order boundary value problems. Appl Math Comput. 2007;188:1437–44. Available from: https://doi.org/10.1016/j.amc.2006.11.015
El-Gamel M. A comparison between the Sinc-Galerkin and the modified decomposition methods for solving two-point boundary value problems. J Comput Phys. 2007;223:369–83. Available from: https://doi.org/10.1016/j.jcp.2006.09.025
Ibikli E. Approximation by Bernstein-Chlodowsky polynomials. Hacettepe J Math Stat. 2003;32:1-5. Available from: https://dergipark.org.tr/en/download/article-file/1170991
Bykiazic I. Approximation by Stancu-Chlodowsky polynomials. Comput Math Appl. 2010;59:274–82. Available from: https://doi.org/10.1016/j.camwa.2009.07.054