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

Eid Doha 1 , Ali Bhrawy, and Samer Ezz-Eldien 4
  • 1 Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt
  • 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
  • 3 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, 62511, Egypt
  • 4 Department of Basic Science, Institute of Information Technology, Modern Academy, Cairo, 11931, Egypt

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.

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  • [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)

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

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

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

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

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

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

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

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

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

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

  • [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.2011.06.016

  • [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.2011.05.011

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

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

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

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

  • [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.2011.07.024

  • [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.2011.12.031

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

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

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

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

  • [24] F. I. Taukenova, M. Kh. Shkhanukov-Lafishev, Comput. Math. Math. Phys. 46, 1785 (2006) http://dx.doi.org/10.1134/S0965542506100149

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

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

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

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

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

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

  • [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/10.1007/978-1-4020-6042-7

  • [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)00255-7

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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