Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Tbilisi Mathematical Journal

Editor-in-Chief: Inassaridze, Hvedri

2 Issues per year


Mathematical Citation Quotient (MCQ) 2016: 0.14

Open Access
Online
ISSN
1512-0139
See all formats and pricing
More options …

Cubic B-spline collocation method for solving time fractional gas dynamics equation

A. Esen
  • Corresponding author
  • Department of Mathematics, Faculty of Science and Art, Inonu University, Malatya, 44280, Turkey
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ O. Tasbozan
  • Corresponding author
  • Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Hatay, 31000, Turkey
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2015-10-16 | DOI: https://doi.org/10.1515/tmj-2015-0024

Abstract

In the present manuscript, a cubic B-spline finite element collocation method has been used to obtain numerical solutions of the nonlinear time fractional gas dynamics equation. While the Caputo form is used for the time fractional derivative appearing in the equation, the L1 discretization formula is applied to the equation in terms of time. It has been seen that the results of the present study are in agreement with the those of exact solution. Therefore, the present method can be used as an alternative and efficient one to find out the numerical solutions of both linear and nonlinear fractional differential equations available in the literature. In order to control the accuracy and efficiency of the present method, the error norms L2 and L1 have been calculated. It is evident that the newly obtained numerical solutions by the present method can be computed easily with the implementation and effectiveness of the approach used in the article.

Keywords: Finite element method; collocation; time fractional gas dynamics equation; cubic B-spline

References

  • [1] K. B. Oldham and J. Spanier, The fractional calculus, Academic, New York, 1974. Google Scholar

  • [2] J. Singh, D. Kumar and A. Kilicman, Homotopy perturbation method for frac- tional gas dynamics equation using Sumudu transform, Abstr. Appl. Anal. (2013), http://dx.doi.org/10.1155/2013/934060, Article ID 934060, 8 pp. CrossrefGoogle Scholar

  • [3] L. Debnath, Fractional integral and fractional differential equations in uid mechan- ics, Fract. Calc. Appl. Anal. 6 (2003) 119-155. Google Scholar

  • [4] S. Monami and Z. Odibat, Analytical approach to linear fractional partial differential equations arising in uid mechanics, Phys. Lett. A 355 (2006) 271-279. Google Scholar

  • [5] D.L. Logan, A first course in the finite element method (Fourth Edition), Thomson, 2007. Google Scholar

  • [6] A. A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006. Google Scholar

  • [7] S. S. Ray, Exact solutions for time-fractional diffusion-wave equations by decomposi- tion method, Phys. Scr. 75 (2007) 53-61. Web of ScienceGoogle Scholar

  • [8] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlin. Dynam. 29 (2002) 145-155. CrossrefGoogle Scholar

  • [9] H. Jafari and S.Momani, Solving fractional diffusion and waves equations by modifiy- ing homotopy perturbation method, Phys. Lett. A 370 (2007) 388-396. Google Scholar

  • [10] A. Esen, Y. Ucar, N. Yagmurlu and O. Tasbozan, A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations, Math. Model. and Anal. 18 (2013) 260-273. Google Scholar

  • [11] A. Esen, O. Tasbozan, Y. Ucar and N.M. Yagmurlu, A B-spline collocation method for solving fractional diffusion and fractional diffusion-wave equations, Tbilisi Math- ematical Journal 8 (2015) 181-193. Google Scholar

  • [12] A. Mohebbi, A. Mostafa and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrodinger equation arising in quantum mechanics, Eng. Anal. with Bound. Elem. 37 (2013) 475-485. Web of ScienceCrossrefGoogle Scholar

  • [13] V. R. Hosseini, W. Chen and Z. Avazzade, Numerical solution of fractional telegraph equation by using radial basis functions, Eng. Anal. with Bound. Elem. 38 (2014) 31-39. Web of ScienceCrossrefGoogle Scholar

  • [14] L.Wei, H. Dai, D. Zhang and Z. Si, Fully discrete local discontinuous Galerkin method for solving the fractional telegraph equation, Calcolo 51 (2014) 175-192. Web of ScienceGoogle Scholar

  • [15] J. Q. Murillo and S.B. Yuste, An explicit difference method for solving fractional diffusion and diffusion-wave equations in the Caputo form, J. Comput. Nonlinear Dynam. 6 (2011) 021014. Google Scholar

  • [16] S. Das and R. Kumar, Approximate analytical solutions of fractional gas dynamic equations, Appl. Math. and Comput. 217 (2011) 9905-9915. Google Scholar

  • [17] T-P. Liu, Nonlinear waves in mechanics and gas dynamics, Defense Technical Infor- mation Center Accession Number: ADA 238340 (1990). Google Scholar

  • [18] M. Rasulov and T. Karaguler, Finite difference schemes for solving system equations of gas dynamic ina class of discontinuous functions, Appl. Math. and Comput. 143 (2003) 45-164. Google Scholar

  • [19] I. Podlubny, Fractional differential dquations, Academic Press, San Diego, 1999. Google Scholar

  • [20] P. M. Prenter, Splines and variasyonel methods, New York, John Wiley, 1975. Google Scholar

About the article

Published Online: 2015-10-16


Citation Information: Tbilisi Mathematical Journal, Volume 8, Issue 2, ISSN (Online) 1512-0139, DOI: https://doi.org/10.1515/tmj-2015-0024.

Export Citation

© 2015 A. Esen, O. Tasbozan. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in