Mittag-Leffler-Pade approximations for the numerical solution of space and time fractional diffusion equations

 
 
 
  • Abstract
  • Keywords
  • References
  • PDF
  • Abstract


    Anomalous diffusion and non-exponential relaxation patterns can be described by a space - time fractional diffusion equation. This paper aims to present a Pade approximation for Mittag-Leffler function mixed finite difference method to develop a numerical method to obtain an approximate solution for the space and time fractional diffusion equation. The truncation error of the method is theoretically analyzed. It is proved that the numerical proposed method is unconditionally stable from the matrix analysis point of view. Finally, some numerical results are given, which demonstrate the efficiency of the approximate scheme.


  • Keywords


    Fractional diffusion equation, Pade approximation, finite difference method, Mittag-Leffler function, stability and convergence.

  • References


      [1] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

      [2] F. M. Atici, P. W. Eloe, Gronwall’s inequality on discrete fractional calculus, Computers and Mathematics with Applications, 64 (2012) 3193–3200.

      [3] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

      [4] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

      [5] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis, 5 (2002) 367–386.

      [6] D. A. Murio, Implicit finite difference approximation for time fractional diffusion equations, Computers and Mathematics with Applications, 56 (2008) 1138–1145.

      [7] M. M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics, 172 (2004) 65–77.

      [8] M. M. Meerschaert, C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Applied Numerical Mathematics, 56 (2006) 80–90.

      [9] Q. Yang, F. Liu, I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Mathematical Modelling, 34 (2010) 200–218.

      [10] N.H. Sweilam, M.M. Khader, A.M. Nagy, Numerical solution of two-sided space-fractional wave equation using finite difference method, Journal of Computational and Applied Mathematics, 235 (2011) 2832–2841.

      [11] N. H. Swellam, M. M. Khader, A. M. S. Mahdy, Crank-Nicolson finite difference method for solving time-fractional diffusion equation, Journal of Fractional Calculus and Applications, 2 (2012) 1–9.

      [12] E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics, 228 (2009) 4038–4054.

      [13] F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Applied Mathematical Modelling, 38 (2014) 3871–3878.

      [14] H. Hejazi, T. Moroney, F. Liu, Stability and convergence of a finite volume method for the space fractional advection-dispersion equation, Journal of Computational and Applied Mathematics, 255 (2014) 684–697.

      [15] A. Elsaid, The variational iteration method for solving Riesz fractional partial differential equations, Computers and Mathematics with Applications, 60 (2010) 1940–1947.

      [16] V. Turut, N. Guzel, On solving partial differential equations of fractional order by using the variational iteration method and multivariate Pade approximations, European Journal of Pure and Applied Mathematics, 6 (2013) 147–171.

      [17] E. Sousa, Numerical approximations for fractional diffusion equations via splines, Computers and Mathematics with Applications, 62 (2011) 938–944.

      [18] T. S. El-Danaf, A. R. Hadhoud, Parametric spline functions for the solution of the one time fractional Burgers equation, Applied Mathematical Modelling, 36 (2012) 4557–4564.

      [19] S. K. Vanani, A. Aminataei, Tau approximate solution of fractional partial differential equations, Computers and Mathematics with Applications, 62 (2011) 1075–1083.

      [20] A. Cetinkaya, O. Kiymaz, The solution of the time-fractional diffusion equation by the generalized differential transform method, Mathematical and Computer Modelling, 57 (2013) 2349–2354.

      [21] Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters, 21 (2008) 194–199.

      [22] Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Applied Mathematics and Computation, 182 (2006) 1048–1055.

      [23] S. S. Ray, Analytical solution for the space fractional diffusion equation by two-step Adomian decomposition method, Communications in Nonlinear Science and Numerical Simulation, 14 (2009) 1295–1306.

      [24] M. Garg, A. Sharma, Solution of space-time fractional telegraph equation by Adomian decomposition method, Journal of Inequalities and Special Functions, 2 (2011) 1–7.

      [25] E. Doha, A. Bhrawy, S. Ezz-Eldien, Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Applied Mathematical Modelling, 35 (2011) 5662–5672.

      [26] A. Borhanifar, S. Valizadeh, Numerical solution for fractional partial differential equations using Crank-Nicolson method with Grunwald estimate, Walailak Journal of Science and Technology, 9 (2012) 433–444.

      [27] F. Mainardi, R. Gorenflo, On Mittag-Leffler-type functions in fractional evolution processes, Journal of Computational and Applied Mathematics, 118 (2000) 283–299.

      [28] M. Yousuf, On the class of high order time stepping schemes based on Pad´e approximations for the numerical solution of Burgers equation, Applied Mathematics and Computation, 205 (2008) 442–453.

      [29] R. L. Burden, J. D. Faires, Numerical Analysis, Brooks/Cole Pub, Boston, 1997.

      [30] E. Isaacson, H. Keller, Analysis of Numerical Methods, Wiley, New York, 1966.


 

View

Download

Article ID: 4340
 
DOI: 10.14419/ijamr.v4i4.4340




Copyright © 2012-2015 Science Publishing Corporation Inc. All rights reserved.