Research Article
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Year 2021, Volume: 9 Issue: 1, 28 - 35, 01.03.2021
https://doi.org/10.36753/mathenot.614732

Abstract

References

  • [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).

Bernstein Operator Approach for Solving Linear Differential Equations

Year 2021, Volume: 9 Issue: 1, 28 - 35, 01.03.2021
https://doi.org/10.36753/mathenot.614732

Abstract

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.

References

  • [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).
There are 39 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Neşe İşler Acar 0000-0003-3894-5950

Publication Date March 1, 2021
Submission Date September 3, 2019
Acceptance Date October 9, 2020
Published in Issue Year 2021 Volume: 9 Issue: 1

Cite

APA İşler Acar, N. (2021). Bernstein Operator Approach for Solving Linear Differential Equations. Mathematical Sciences and Applications E-Notes, 9(1), 28-35. https://doi.org/10.36753/mathenot.614732
AMA İşler Acar N. Bernstein Operator Approach for Solving Linear Differential Equations. Math. Sci. Appl. E-Notes. March 2021;9(1):28-35. doi:10.36753/mathenot.614732
Chicago İşler Acar, Neşe. “Bernstein Operator Approach for Solving Linear Differential Equations”. Mathematical Sciences and Applications E-Notes 9, no. 1 (March 2021): 28-35. https://doi.org/10.36753/mathenot.614732.
EndNote İşler Acar N (March 1, 2021) Bernstein Operator Approach for Solving Linear Differential Equations. Mathematical Sciences and Applications E-Notes 9 1 28–35.
IEEE N. İşler Acar, “Bernstein Operator Approach for Solving Linear Differential Equations”, Math. Sci. Appl. E-Notes, vol. 9, no. 1, pp. 28–35, 2021, doi: 10.36753/mathenot.614732.
ISNAD İşler Acar, Neşe. “Bernstein Operator Approach for Solving Linear Differential Equations”. Mathematical Sciences and Applications E-Notes 9/1 (March 2021), 28-35. https://doi.org/10.36753/mathenot.614732.
JAMA İşler Acar N. Bernstein Operator Approach for Solving Linear Differential Equations. Math. Sci. Appl. E-Notes. 2021;9:28–35.
MLA İşler Acar, Neşe. “Bernstein Operator Approach for Solving Linear Differential Equations”. Mathematical Sciences and Applications E-Notes, vol. 9, no. 1, 2021, pp. 28-35, doi:10.36753/mathenot.614732.
Vancouver İşler Acar N. Bernstein Operator Approach for Solving Linear Differential Equations. Math. Sci. Appl. E-Notes. 2021;9(1):28-35.

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