Bernstein Operator Approach for Solving Linear Differential Equations
In this study, an alternative numerical method having regard to the Bernstein operator is generated for approximate solutions of linear differential equations in the most general form under the initial and boundary conditions. Some applications are also revealed to show how the procedure can be performed for the problems.
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- [1] Akyüz-Daşcıoğlu, A., Işler, N.: Bernstein Collocation Method for Solving Nonlinear Differential Equations. Mathematical
and Computational Applications. 18 (3), 293-300 (2013).
- [2] Akyüz-Daşcıoğlu, A., Acar Işler, N.: Bernstein Collocation Method for Solving Linear Differential Equations. Gazi
University Journal of Science. 26 (4), 527-534 (2013).
- [3] Bataineh, A., Isik, O., Aloushoush, N., Shawagfeh, N.: Bernstein Operational Matrix with Error Analysis for Solving
High Order Delay Differential Equations. Int. J. Appl. Comput. Math. 3, 1749-1762 (2017).
- [4] Bhatta, D.D., Bhatti, M.I.: Numerical Solution of KdV equation using modified Bernstein polynomials. Applied
Mathematics and Computation. 174 (2), 1255-1268 (2006).
- [5] Bhattacharya, S., Mandal, B.N.: Use of Bernstein polynomials in numerical solutions of Volterra integral equations.
Applied Mathematical Sciences. 2, 1773-1787 (2008).
- [6] Bhattacharya, S., Mandal, B.N.: Numerical solution of a singular integro-differential equation. Applied Mathematics
and Computation. 195, 346-350 (2008).
- [7] Bhatti, M.I., Bracken, P.: Solutions of differential equations in a Bernstein polynomial basis. Journal of Computational
and Applied Mathematics. 205 (2), 172-280 (2007).
- [8] Bernstein, S.: Démonstration du théorème de Weierstrass Fondeé sur le calcul des probabilités . Commun. Soc. Math.
Kharkow. 13 (2), 1-2 (1912).
- [9] Büyükyazıcı, İ: Approximation by Stancu-Chlodowsky polynomials. Computers and Mathematics with Applications.
59 (1), 274-282 (2010).
- [10] Doha, E.H., Bhrawy, A.H., Saker, M.A.: On the derivatives of Bernstein polynomials: An application for the solution of
high even-order differential equations. Boundary Value Problems. 2011, 1-16 (2011).
- [11] Doha, E.H., Bhrawy, A.H., Saker, M.A.: Integrals of Bernstein polynomials: An application for the solution of high
even-order differential equations. Applied Mathematics Letters. 24, 559-565 (2011).
- [12] El-Gamel, M.: A comparison between the Sinc-Galerkin and the modified decomposition methods for solving two-point
boundary-value problems. Journal of Computational Pysics. 223 (1), 369-383 (2007).
- [13] Faheem, K., Ghulam, M., Muhammad, O., Haziqa, K.: Numerical approach based on Bernstein polynomials for solving
mixed Volterra-Fredholm integral equations. AIP Advances. 7, 1-14 (2017). https://doi.org/10.1063/1.5008818
- [14] Farouki, R.T., Rajan, V.T.: Algorithms for polynomials in Bernstein form. Computer Aided Geometric Design. 5,
1-26 (1988).
- [15] Frammartino, C.: A Nyström method for solving a boundary value problem on [-1; 1]. Calcolo. 47, 1-19 (2010).
- [16] Golbabai, A., Javidi, M.: Application of homotopy perturbation method for solving eighth-order boundary value problems.
Applied Mathematics and Computation. 191 (2), 334-346 (2007).
- [17] Hesameddini, E., Khorramizadeh, M., Shahbazi, M.: Bernstein polynomials method for solving Volterra-Fredholm
integral equations. Bull. Math. Soc. Sci. Math. Roumanie Tome 60. 108 (1), 59-68 (2017).
- [18] I¸sık, O.R., Sezer, M., Güney, Z.: A Rational approximation based on Bernstein polynomials for high order initial and
boundary value problems. Applied Mathematics and Computation. 217 (22), 9438-9450 (2011).
- [19] ˙I¸sler Acar, N., Da¸scıo ˘ glu, A.: A projection method for linear Fredholm-Volterra integro-differential equations. Journal
of Taibah University for Science. 13 (1), 644-650 (2019).
- [20] Jani, M., Babolian, E., Javadi, S.: Bernstein modal basis: Application to the spectral Petrov-Galerkin method for
fractional partial differential equations. Math Meth Appl Sci. 40, 7663–7672 (2017).
- [21] Javadi, S., Babolian, E., Tahari, Z.: Solving generalized pantograph equations by shifted orthonormal Bernstein
polynomials. Journal of Computational and Applied Mathematics. 303, 1-14 (2016).
- [22] Kadkhoda, N.: A numerical approach for solving variable order differential equations using Bernstein polynomials.
Alexandria Engineering Journal. (2020). https://doi.org/10.1016/j.aej.2020.05.009
- [23] Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions. Dokl. Akad. Nauk
SSSR (N.S). 90, 961-964 (1953).
- [24] Lorentz, G.G.: Bernstein polynomials. Chelsea Publishing. New York (1986).
- [25] Mirzaee, F., Samadyar, N.: Parameters estimation of HIV infection model of CD4+ T-cells by applying orthonormal
Bernstein collocation method. International Journal of Biomathematics. 11 (2), 1-19 (2018).
- [26] Mohamadi, M., Babolian, E., Yousefi, S.A.: A Solution For Volterra Integral Equations of the First Kind Based on
Bernstein Polynomials. Int. J. Industrial Mathematics. 10 (1), 1-9 (2018).
- [27] Ordokhani, Y., Davaei far, S.: Approximate solutions of differential equations by using the Bernstein polynomials.
International Scholarly Research Network ISRN Applied Mathematics. 2011 (1), 1-15 (2011).
- [28] Parand, K., Sayyed, A., Hossayni, J.A.R.: Operation matrix method based on Bernstein polynomials for the Riccati
differential equation and Volterra population model. Applied Mathematical Modelling. 40, 993-1011 (2016).
- [29] Pirabaharan P., Chandrakumar, R.D.: A computational method for solving a class of singular boundary value problems
arising in science and engineering. Egyptian journal of basic and applied sciences. 3, 383-391 (2016).
- [30] Quasim, A.F., Hamed, A.A.: Treating Transcendental Functions in Partial Differential Equations Using the Variational
Iteration Method with Bernstein Polynomials. Hindawi International Journal of Mathematics and Mathematical
Sciences. 2019, 1-8 (2019). https://doi.org/10.1155/2019/2872867
- [31] Quasim, A.F., Al-Ravi, E.S.: Adomian Decomposition Method with Modified Bernstein Polynomials for Solving
Ordinary and Partial Differential Equations. Hindawi Journal of Applied Mathematics. 2018, 1-9 (2018).
https://doi.org/10.1155/2018/1803107
- [32] Ramadan, M.A., Lashien, I.F., Zahra,W.K.: High order accuracy nonpolynomial spline solutions for 2 th order two
point boundary value problems. Applied Mathematics and Computation. 204 (2), 920-927 (2008).
- [33] Rani, D., Mishra, V.: Approximate Solution of Boundary Value Problem with Bernstein Polynomial Laplace Decomposition
Method. International Journal of Pure and Applied Mathematics. 114 (4), 823-833 (2017).
- [34] Saadatmandi, A.: Bernstein operational matrix of fractional derivatives and its applications. Applied Mathematical
Modelling. 38, 1365-1372 (2014).
- [35] Shirin, A., Islam, A.S.: Numerical solutions of Fredholm integral equations using Bernstein polynomials. Journal of
Scientific Research. 2 (2), 264-272 (2010).
- [36] Siddiqi, S.S., Akram, G.: Septic spline solutions of sixth-order boundary value problems. Journal of Computational
and Applied Mathematics. 215 (1), 288-301 (2008).
- [37] ¸Suayip, Y.: A collocation method based on Bernstein polynomials to solve nonlinear Fredholm–Volterra integro-differential
equations. Applied Mathematics and Computation. 273, 142-154 (2016).
- [38] Yi-ming, C., Li-qing, L., Dayan, L., Driss, B.: Numerical study of a class of variable order nonlinear fractional
differential equation in terms of Bernstein polynomials. Ain Shams Engineering Journal. 9, 1235-1241 (2018).
- [39] Yousefi, S.A., Dehghan, B.M.: Bernstein Ritz-Galerkin method for solving an initial-boundary value problem that
combines Neumann and integral condition for the wave equation. Numerical Methods for Partial Differantial
Equations. 26, 1236-1246 (2009).