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

Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Feng, Jonathan

Managing Editor: Lesna-Szreter, Paulina

1 Issue per year


IMPACT FACTOR 2016 (Open Physics): 0.745
IMPACT FACTOR 2016 (Central European Journal of Physics): 0.765

CiteScore 2016: 0.82

SCImago Journal Rank (SJR) 2015: 0.458
Source Normalized Impact per Paper (SNIP) 2015: 1.142

Open Access
Online
ISSN
2391-5471
See all formats and pricing
More options …
Volume 11, Issue 10 (Oct 2013)

Issues

Numerical approximations for fractional diffusion equations via a Chebyshev spectral-tau method

Eid Doha / Ali Bhrawy
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
  • Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, 62511, Egypt
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Samer Ezz-Eldien
Published Online: 2013-12-19 | DOI: https://doi.org/10.2478/s11534-013-0264-7

Abstract

In this paper, a class of fractional diffusion equations with variable coefficients is considered. An accurate and efficient spectral tau technique for solving the fractional diffusion equations numerically is proposed. This method is based upon Chebyshev tau approximation together with Chebyshev operational matrix of Caputo fractional differentiation. Such approach has the advantage of reducing the problem to the solution of a system of algebraic equations, which may then be solved by any standard numerical technique. We apply this general method to solve four specific examples. In each of the examples considered, the numerical results show that the proposed method is of high accuracy and is efficient for solving the time-dependent fractional diffusion equations.

Keywords: fractional diffusion equations; spectral tau technique; operational matrix; shifted Chebyshev polynomials

  • [1] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional calculus models and numerical methods (Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, New York, 2012) Google Scholar

  • [2] D. Baleanu, J.H. Asad, I. Petras, Romanian Reports on Physics 64, 907 (2012) Google Scholar

  • [3] M. Dalir, M. Bashour, Applied Mathematical Sciences 4, 1021 (2010) Google Scholar

  • [4] S. Das, Functional Fractional Calculus for System Identification and Controls (Springer, New York, 2008) Google Scholar

  • [5] I. Podlubny, Fractional Differential Equations (Academic Press Inc., San Diego, CA, 1999) Google Scholar

  • [6] H. Jafari, M. Saeidy, D. Baleanu, Cent. Eur. J. Phys. 10, 76 (2012) http://dx.doi.org/10.2478/s11534-011-0083-7CrossrefGoogle Scholar

  • [7] J. Deng, L. Ma, Appl. Math. Lett. 23, 676 (2010) http://dx.doi.org/10.1016/j.aml.2010.02.007CrossrefGoogle Scholar

  • [8] A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Elsevier, San Diego, 2006) Google Scholar

  • [9] C. G. Li, M. Kosti, M. Li, Sergey Piskarev, Fract. Calc. Appl. Anal. 15, 639 (2012) Google Scholar

  • [10] S Salahshour, T Allahviranloo, S Abbasbandy, D Baleanu, Advan. Diff. Equ. 2012, 112 (2012) http://dx.doi.org/10.1186/1687-1847-2012-112CrossrefGoogle Scholar

  • [11] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics (Springer-Verlag, New York, 1989) Google Scholar

  • [12] A. H. Bhrawy, A.S. Alofi, S.S. Ezz-Eldien, Appl. Math. Lett. 24, 2146 (2011) http://dx.doi.org/10.1016/j.aml.2011.06.016CrossrefGoogle Scholar

  • [13] E. H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 35, 5662 (2011) http://dx.doi.org/10.1016/j.apm.2011.05.011CrossrefGoogle Scholar

  • [14] A. H. Bhrawy, M.M. Al-Shomrani, Advances in Difference Equations 2012, 8 (2012) http://dx.doi.org/10.1186/1687-1847-2012-8CrossrefGoogle Scholar

  • [15] C. Li, F. Zeng, F. Liu, Fractional Calculus and Applied Analysis 15, 383 (2012) http://dx.doi.org/10.2478/s13540-012-0028-xCrossrefGoogle Scholar

  • [16] F. Ghoreishi, S. Yazdani, Comput. Math. Appl. 61, 30 (2011) http://dx.doi.org/10.1016/j.camwa.2010.10.027CrossrefGoogle Scholar

  • [17] A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 59, 1326 (2010) http://dx.doi.org/10.1016/j.camwa.2009.07.006CrossrefGoogle Scholar

  • [18] E. H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Comput. Math. Appl. 62, 2364 (2011) http://dx.doi.org/10.1016/j.camwa.2011.07.024CrossrefGoogle Scholar

  • [19] E. H. Doha, A.H. Bhrawy, S.S. Ezz-Eldien, Appl. Math. Model. 36, 4931 (2012) http://dx.doi.org/10.1016/j.apm.2011.12.031CrossrefGoogle Scholar

  • [20] A. H. Bhrawy, M.M. Tharwat, A. Yildirim, Appl. Math. Modell. 37, 4245 (2013) http://dx.doi.org/10.1016/j.apm.2012.08.022CrossrefGoogle Scholar

  • [21] A. H. Bhrawy, A.S. Alofi, Appl. Math. Lett. 26, 25 (2013) http://dx.doi.org/10.1016/j.aml.2012.01.027CrossrefGoogle Scholar

  • [22] A. H. Bhrawy, M.A. Alghamdi, T.M. Taha, Advances in Difference Equations 2012, 179 (2012) http://dx.doi.org/10.1186/1687-1847-2012-179CrossrefGoogle Scholar

  • [23] K. Diethelm, N.J. Ford, BIT 42, 490 (2002) Google Scholar

  • [24] F. I. Taukenova, M. Kh. Shkhanukov-Lafishev, Comput. Math. Math. Phys. 46, 1785 (2006) http://dx.doi.org/10.1134/S0965542506100149CrossrefGoogle Scholar

  • [25] S. Karimi Vanani, A. Aminataei, Comput. Math. Appl. 62, 1075 (2011) http://dx.doi.org/10.1016/j.camwa.2011.03.013CrossrefGoogle Scholar

  • [26] X. C. Li, W. Chen, The European Physical Journal Special Topics 193, 221 (2011) http://dx.doi.org/10.1140/epjst/e2011-01393-3CrossrefGoogle Scholar

  • [27] A. K. Golmankhaneh, T. Khatuni, N. A. Porghoveh, D. Baleanu, Cent. Eur. J. Phys. 10, 966 (2012) http://dx.doi.org/10.2478/s11534-012-0038-7CrossrefGoogle Scholar

  • [28] O. P. Agrawal, Nonlinear Dynamics 29, 145 (2002) http://dx.doi.org/10.1023/A:1016539022492CrossrefGoogle Scholar

  • [29] R. L. Magin, Fractional Calculus in Bioengineering (Begell House Publisher., Inc., Connecticut, 2006) Google Scholar

  • [30] R. Metzler, J. Klafter, Physics Reports 339, 1 (2000) http://dx.doi.org/10.1016/S0370-1573(00)00070-3CrossrefGoogle Scholar

  • [31] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (Springer, Netherlands, 2007) http://dx.doi.org/10.1007/978-1-4020-6042-7CrossrefGoogle Scholar

  • [32] E. Scalas, R. Gorenflo, F. Mainardi, Physica A: Statistical Mechanics and its Applications 284, 376 (2000) http://dx.doi.org/10.1016/S0378-4371(00)00255-7CrossrefGoogle Scholar

  • [33] C. Tadjeran, M.M. Meerschaert, H.-P. Scheffler, J. Comput. Phys. 213, 205 (2006) http://dx.doi.org/10.1016/j.jcp.2005.08.008CrossrefGoogle Scholar

  • [34] M. Cui, J. Comput. Phys. 228, 7792 (2009) http://dx.doi.org/10.1016/j.jcp.2009.07.021CrossrefGoogle Scholar

  • [35] A. Saadatmandi, M. Dehghan, Comput. Math. Appl. 62, 1135 (2011) http://dx.doi.org/10.1016/j.camwa.2011.04.014CrossrefGoogle Scholar

  • [36] S. Shen, F. Liu, ANZIAM J. 46, 871 (2005) Google Scholar

  • [37] Z. Ding, A. Xiao, M. Li, J. Comput. Appl. Math. 233, 1905 (2010) http://dx.doi.org/10.1016/j.cam.2009.09.027CrossrefGoogle Scholar

  • [38] F. Liu, P. Zhuang, K. Burrage, Comput. Math. Appl. 64, 2990 (2012) http://dx.doi.org/10.1016/j.camwa.2012.01.020CrossrefGoogle Scholar

  • [39] C. Celik, M. Duman, Journal of Computational Physics 231, 1743 (2012) http://dx.doi.org/10.1016/j.jcp.2011.11.008CrossrefGoogle Scholar

  • [40] E. Sousa, J. Comput. Phys. 228, 4038 (2009) http://dx.doi.org/10.1016/j.jcp.2009.02.011CrossrefGoogle Scholar

  • [41] L. Su, W. Wang, Z. Yang, Phys. Lett. A 373, 4405 (2009) http://dx.doi.org/10.1016/j.physleta.2009.10.004CrossrefGoogle Scholar

  • [42] E. L. Ortiz, The tau method, SIAM J. Numer. Anal. Optim. 12, 480 (1969) http://dx.doi.org/10.1137/0706044CrossrefGoogle Scholar

  • [43] E. L. Ortiz, H. Samara, Comput. Math. Appl. 10, 5 (1984) http://dx.doi.org/10.1016/0898-1221(84)90081-6CrossrefGoogle Scholar

  • [44] M. Caputo, J. Roy Austral. Soc. 13, 529 (1967) http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.xCrossrefGoogle Scholar

  • [45] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley & Sons Inc., New York, 1993) Google Scholar

  • [46] E. Sousa, Comput. Math. Appl. 62, 938 (2011) http://dx.doi.org/10.1016/j.camwa.2011.04.015CrossrefGoogle Scholar

About the article

Published Online: 2013-12-19

Published in Print: 2013-10-01


Citation Information: Open Physics, ISSN (Online) 2391-5471, DOI: https://doi.org/10.2478/s11534-013-0264-7.

Export Citation

© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Yuankui Ma and Xingxing Lv
Mathematical Problems in Engineering, 2017, Volume 2017, Page 1
[2]
Tingting Wang and Han Zhang
Mathematical Problems in Engineering, 2015, Volume 2015, Page 1
[3]
Xiaoxue Li
Mathematical Problems in Engineering, 2015, Volume 2015, Page 1
[4]
A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien, and Robert A. Van Gorder
The European Physical Journal Plus, 2014, Volume 129, Number 12
[5]
Eid H Doha, Ali H Bhrawy, Dumitru Baleanu, and Samer S Ezz-Eldien
Advances in Difference Equations, 2014, Volume 2014, Number 1, Page 231
[6]
Eid Doha, Ali Bhrawy, and Mohammed Abdelkawy
Open Physics, 2014, Volume 12, Number 9
[7]
[8]
A.H. Bhrawy, E.H. Doha, D. Baleanu, and S.S. Ezz-Eldien
Journal of Computational Physics, 2015, Volume 293, Page 142

Comments (0)

Please log in or register to comment.
Log in