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Journals / Gazi University Journal of Science / 2013 / Volume: 26 Issue: 4

Bernstein Collocation Method for Solving Linear Differential Equations

Ayşegül Daşçıoğlu
Neşe Acar
Pages
—
Publication Date
2014-01-02
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

Some numerical solutions are given to illustrate the accuracy, efficiency and implementation of the method, and the results of the proposed method are also compared with the other methods in several examples.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this study, a new collocation method based on Bernstein polynomials defined on the interval [a, b] is introduced for approximate solutions of initial and boundary value problems involving higher order linear differential equations with variable coefficients.
📊
Key Findings & Data:Error analysis of the method is demonstrated.
💡
Conclusion & Impact:Some numerical solutions are given to illustrate the accuracy, efficiency and implementation of the method, and the results of the proposed method are also compared with the other methods in several examples.
🔬Design:Conference Paper
📍Scope:Comprehensive Sample
⚙️Variables:Bernstein polynomial approximation · Linear differential equations · Collocation method

Abstract

In this study, a new collocation method based on Bernstein polynomials defined on the interval [a, b] is introduced for approximate solutions of initial and boundary value problems involving higher order linear differential equations with variable coefficients. Error analysis of the method is demonstrated. Some numerical solutions are given to illustrate the accuracy, efficiency and implementation of the method, and the results of the proposed method are also compared with the other methods in several examples.

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Cite This Article

Daşçıoğlu, A., Acar, N. (2014). Bernstein Collocation Method for Solving Linear Differential Equations. Gazi University Journal of Science, Volume: 26 Issue: 4. https://ajindex.com/articles/1269706

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References (18)

  1. Farouki, R.T. and Rajan, V.T., Algorithms for polynomials in Bernstein form, Computer Aided Geometric Design, 5: 1-26, (1988).
  2. Farin, G., Curves and surfaces for CAGD fifth edition, Academic Press, United States of America, (2002).
  3. Bhatti, M.I. and Bracken, P., Solutions of differential equations in a Bernstein polynomial basis, Journal of Computational and Applied Mathematics, 205: 272-280, (2007).
  4. Doha, E.H., Bhrawy, A.H. and 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).
  5. Doha, E. H., Bhrawy, A. H. and 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).
  6. Işık, O.R., Sezer, M. and Güney, Z., A Rational approximation based on Bernstein polynomials for high order initial and boundary value problems, Applied Mathematics and Computation, 217: 9438-9450, (2011).
  7. Işık, O.R., Güney, Z. and Sezer, M., Bernstein series solutions of pantograph equations using polynomial interpolation, Journal of Difference Equations and Applications, 18: 357-374, (2010).
  8. Ordokhani, Y. and Davaei far, S., Approximate solutions of differential equations by using the Bernstein polynomials, International Scholarly Research Network ISRN Applied Mathematics, 2011: 1-15, (2011).
  9. Bhattacharya, S. and Mandal, B.N., Use of Bernstein polynomials in numerical solutions of Volterra integral equations, Applied Mathematical Sciences, 2: 1773-1787, (2008).
  10. Shirin, A. and Islam, A.S., Numerical solutions of Fredholm integral equations using Bernstein polynomials, Journal of Scientific Research, 2: 264-272, (2010).
  11. Bhatta, D.D. and Bhatti, M.I, Numerical Solution of KdV equation using modified Bernstein polynomials, Computation, 174: 1255-1268, (2006). Mathematics and
  12. Stewart, G.W., Matrix algorithms volume I: Basic decompositions, SIAM, Philadelphia, (1998).
  13. Yousefi, S.A. and 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 Differential Equations, 26: 1236-1246, (2009).
  14. Bhattacharya, S. and Mandal, B.N., Numerical solution of a singular integro- differential equation, Applied Mathematics and Computation, 195: 346- 350, (2008).
  15. Lorentz, G.G., Bernstein polynomials, Chelsea Publishing, New York, N.Y., (1986).
  16. Rivlin, T.J., An introduction to the approximation of functions, Dover Publications, N.Y., (2003).
  17. Siddiqi, S.S. and Akram, G., Septic spline solutions of sixth-order boundary value problems, Journal Mathematics, 215: 288-301, (2008). and Applied
  18. El-Gamel, M., Cannon, J.R. and Zayed, A.I., Sinc- Galerkin method for solving linear sixth-order boundary value problems, Mathematics and Computation, 73: 1325-1343, (2003).