ANZIAM J. 46(E) pp.C871--C887, 2005.

Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends

S. Shen

F. Liu

(Received 8 October 2004, revised 28 June 2005)

Abstract

The space fractional diffusion equation (SFDE) is obtained from the classical diffusion equation by replacing the second space derivative by a fractional derivative of order a, 1 < a £ 2 . Numerical methods associated with integer-order differential equation, have been extensively treated. On the other hand, studies of the numerical methods and error estimates of fractional order differential equations are quite limited to date. Here, we propose an explicit finite difference approximation (EFDA) for SFDE. An error analysis of the explicit numerical method for SFDE with insulated ends is discussed. We derive the scaling restriction of the stability and convergence of the explicit numerical method. Finally, some numerical results show the diffusion behaviour according to the order of space-fractional derivative and demonstrate that our EFDA is computationally simple for SFDE.

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Authors

S. Shen
F. Liu
School of Mathematical Sciences, Xiamen University, Xiamen 361005, China. mailto:fwliu@xmu.edu.cn.
School of Mathematical Sciences, Queensland University of Technology, Queensland 4001, Australia. mailto:f.liu@qut.edu.au.

Published September 1, 2005. ISSN 1446-8735

References

  1. G. J. Fix and J. P. Roop, Least squares finite element solution of a fractional order two-point boundary value problem, Comput. Math. Appl., 48, 2004, 1017--1044.
  2. R. Gorenflo and F. Mainardi, Random walk models for space fractional diffusion processes, Fractional Calculus & Applied Analysis 1, 1998, 167--191.
  3. R. Gorenflo and F. Mainardi, Approximation of Levy--Feller diffusion by random walk, Journal for Analysis and its Applications (ZAA), 18, 1999, 231--246.
  4. J. A. Tenreiro Machado, A probabilistic interpretation of the fractional-order differentiation, Fractional Calculus & Applied Analysis, 6, No.1, 2003, 73--80.
  5. F. Liu, V. Anh and I. Turner, Numerical solution of the fractional-order Advection-Dispersion Equation, The Procceding of An International Conference on Boundary and Interior Layers-Computational and Asymptotic Methods, Perth, Australia, 2002, 159--164.
  6. F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker--Planck equation, Journal of Computational and Applied Mathematics 166, 2004, 209--219.
  7. F. Liu, V. Anh, I. Turner and P. Zhuang, Numerical simulation for solute transport in fractal porous media, ANZIAM J., 45(E), 2004, C461--C473.
  8. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, Journal of Computational and Applied Mathematics, 172, 2004, 65--77.
  9. P. V. O'Neil, Advanced Engineering Mathematics, Thomson Brooks/Cole, 2003.
  10. I. Podlubny, Fractional Differential Equations, Academic, Press, New York, 1999.
  11. I. Podlubny, Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, Fractional Calculus & Applied Analysis, 5, No.4, 2002, 367--386.
  12. B. West and V. Seshadri, Linear systems with Levy fluctuations, Physica A, 113, 1982, 203--216.
  13. Z. Xu, K. Zhang, Q. Lu, G. Leng, Matrix Theory, Scientific publishing house, 2001.