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Numerical Solutions of the Modified Burgers Equation by a Cubic B-spline Collocation Method

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Abstract

In this paper, a numerical solution of the modified Burgers equation is obtained by a cubic B-spline collocation method. In the solution process, a linearization technique based on quasi-linearization has been applied to deal with the non-linear term appearing in the equation. The computed results are compared with others selected from the available literature. The error norms \(L_{2}\) and \(L_{\infty }\) are computed and found to be sufficiently small. A Fourier stability analysis of the method is also investigated.

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References

  1. Bateman, H.: Some recent researches on the motion of fluids. Mon. Weather Rev. 43, 163–170 (1915)

    Article  Google Scholar 

  2. Mittal, R.C., Jain, R.K.: Numerical solutions of nonlinear Burgers equation with modified cubic B-splines collocation method. Appl. Math. Comput. 218, 7839–7855 (2012)

  3. Benton, E., Platzman, G.W.: A table of folutions of the one-dimensional Burgers equations. Q. Appl. Math. 30, 195–212 (1972)

    MathSciNet  MATH  Google Scholar 

  4. Miller, E.L.: Predictor-corrector studies of Burger’s model of turbulent flow, M. S. Thesis, University of Delaware, Newark (1966)

  5. Dag, I., Irk, D., Sahin, A.: B-spline collocation methods for numerical solutions of the Burgers equation. Math. Probl. Eng. 2005(5), 521–538 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ramadan, M.A., El-Danaf, T.S.: Numerical treatment for the modified burgers equation. Math. Comput. Simul. 70, 90–98 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ramadan, M.A., El-Danaf, T.S., Abd Alaal, F.E.I.: A numerical solution of the Burgers equation using septic B-splines. Chaos, Solitons Fractals 26, 795–804 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Saka, B., Dag, I.: A numerical study of the Burgers equation. J. Frankl. Inst. 345, 328–348 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Irk, D.: Sextic B-spline collocation method for the modified Burgers equation. Kybernetes 38(9), 1599–1620 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Temsah, R.S.: Numerical solutions for convection-diffusion equation using El-Gendi method. Commun. Nonlinear Sci. Numer. Simul. 14, 760–769 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Griewank, A., El-Danaf, T.S.: Efficient accurate numerical treatment of the modified Burgers equation. Appl. Anal. 88(1), 75–87 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bratsos, A.G.: An implicit numerical scheme for the modified Burgers equation, in HERCMA 2009 (9\(^{\text{ o }}\) Hellenic-European conference on computer mathematics and its applications, pp. 24–26 September 2009, Athens

  13. Bratsos, A.G.: A fourth-order numerical scheme for solving the modified Burgers equation. Comput. Math. Appl. 60, 1393–1400 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bratsos, A.G., Petrakis, L.A.: An explicit numerical scheme for the modified Burgers equation. Int. J. Numer. Methods Biomed. Eng. 27, 232–237 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Roshan, T., Bhamra, K.S.: Numerical solutions of the modified Burgers equation by Petrov–Galerkin method. Appl. Math. Comput. 218, 3673–3679 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mittal, R.C., Arora, G.: Numerical solution of the coupled viscous Burgers equation. Commun. Nonlinear Sci. Numer. Simulat. 16, 1304–1313 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Prenter, P.M.: Splines Variational Methods. John Wiley, New York (1975)

    MATH  Google Scholar 

  18. Rubin, S.G., Graves, R.A.: A Cubic spline approximation for problems in fluid mechanics, Nasa TR R-436, Washington, DC, 1975

  19. Burden, R.L., Faires, J.D.: Numerical Analysis. Brooks/cole Cenaage Learning, Boston (2011)

    MATH  Google Scholar 

  20. Smith, G.D.: Numerical Solution of Partial Differential Equation: Finite Difference Method. Learendom Press, Oxford (1978)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, İnönü University, 44280, Malatya, Turkey

    Selcuk Kutluay, Yusuf Ucar & N. Murat Yagmurlu

Authors
  1. Selcuk Kutluay
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  2. Yusuf Ucar
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  3. N. Murat Yagmurlu
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Correspondence to N. Murat Yagmurlu.

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Communicated by Norhashidah M. Ali.

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Kutluay, S., Ucar, Y. & Yagmurlu, N.M. Numerical Solutions of the Modified Burgers Equation by a Cubic B-spline Collocation Method . Bull. Malays. Math. Sci. Soc. 39, 1603–1614 (2016). https://doi.org/10.1007/s40840-015-0262-6

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  • DOI: https://doi.org/10.1007/s40840-015-0262-6

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