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Discrete fractional diffusion equation

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Abstract

The tool of the discrete fractional calculus is introduced to discrete modeling of diffusion problem. A fractional time discretization diffusion model is presented in the Caputo-like delta’s sense. The numerical formula is given in form of the equivalent summation. Then, the diffusion concentration is discussed for various fractional difference orders. The discrete fractional model is a fractionization of the classical difference equation and can be more suitable to depict the random or discrete phenomena compared with fractional partial differential equations.

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References

  1. Hamilton, J.D.: Time series analysis. Princeton University Press, Princeton (1994)

    MATH  Google Scholar 

  2. Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods, and Applications. Marcel Dekker, New York (1992)

  3. Bohner, M., Peterson, A.: Dynamical Equations on Time Scales. Birkhauser, Boston (2001)

    Book  Google Scholar 

  4. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  5. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus Models and Numerical Methods. World Scientific, Singapore (2012)

    MATH  Google Scholar 

  6. Schneider, W., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 7, 1461–1477 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Gorenflo, R., Mainardi, F.: Random walk models for space-fractional diffusion processes. Fract. Calc. Appl. Anal. 1, 167–191 (1998)

    MATH  MathSciNet  Google Scholar 

  9. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gorenflo, R., Mainardi, F., Moretti, D., Paradisi, P.: Time fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129–143 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A 388, 4586–4592 (2009)

    Article  Google Scholar 

  12. Chen, W., Sun, H., Zhang, X., Korosak, D.: Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59, 1754–1758 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. Huang, F., Liu, F.: The time fractional diffusion equation and the advection-dispersion equation. ANZIAM J. 46, 317–330 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Caputo, M.: Linear models of dissipation whose \(Q\) is almost frequency independent-II. Geophys. J. Int. 13, 529–539 (1967)

    Article  Google Scholar 

  15. Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 9–25 (2013)

    Article  MathSciNet  Google Scholar 

  16. Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hristov, J.: Approximate solutions to fractional subdiffusion equations. Eur. Phys. J. Spec. Top. 193, 229–243 (2011)

    Article  Google Scholar 

  18. Machado, J.A.T.: Analysis and design of fractional-order digital control systems. Syst. Anal. Model. Simul. 27, 107–122 (1997)

    MATH  Google Scholar 

  19. Machado, J.A.T.: Discrete-time fractional-order controllers. Fract. Calc. Appl. Anal. 4, 47–66 (2001)

    MATH  MathSciNet  Google Scholar 

  20. Ortigueira, M.D., Matos, C.J.C., Piedade, M.S.: Fractional discrete-time signal processing: scale conversion and linear prediction. Nonlinear Dyn. 29, 173–190 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ortigueira, M.D., Trujillo, J.J.: Generalized Grunwald–Letnikov fractional derivative and its Laplace and Fourier transforms. J. Comput. Nonlinear Dyn. 6, 034501 (2011)

    Article  Google Scholar 

  22. Edelman, M., Tarasov, V.E.: Fractional standard map. Phys. Lett. A 374, 279–285 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  23. Tarasov, V.E., Edelman, M.: Fractional dissipative standard map. Chaos 20, 023127 (2010)

    Article  MathSciNet  Google Scholar 

  24. Pu, Y.F., Zhou, J.L., Yuan, X.: Fractional differential mask: a fractional differential-based approach for multiscale texture enhancement. IEEE Trans. Image Process. 19, 491–511 (2010)

    Article  MathSciNet  Google Scholar 

  25. Atici, F.M., Eloe, P.W.: A transform method in discrete fractional calculus. Int. J. Differ. Equ. 2, 165–176 (2007)

    MathSciNet  Google Scholar 

  26. Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137, 981–989 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  27. Anastassiou, G.A.: About Discrete Fractional Calculus with Inequalities, in: Intelligent Mathematics: Computational Analysis, pp. 575–585. Springer, Berlin (2011)

    Google Scholar 

  28. Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62, 1602–1611 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wu, G.C., Baleanu, D.: Discrete fractional logistic map and its chaos. Nonlinear Dyn. 75, 283–287 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Wu, G.C., Baleanu, D., Zeng, S.D.: Discrete chaos in fractional sine and standard maps. Phys. Lett. A 378, 484–487 (2014)

    Article  MathSciNet  Google Scholar 

  31. Wu, G.C., Baleanu, D.: Chaos synchronization of the discrete fractional logistic map. Signal Process. 102, 96–99 (2014)

    Article  Google Scholar 

  32. Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Discrete-time fractional variational problems. Signal. Process. 91, 513–524 (2011)

    Article  MATH  Google Scholar 

  33. Ferreira, R.A.C., Torres, D.F.M.: Fractional \(h\)-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5, 110–121 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. Mozyrska, D., Girejko, E.: Overview of Fractional \(h\)-Difference Operators, Advances in Harmonic Analysis and Operator Theory. Springer Basel, Berlin (2013)

    Google Scholar 

  35. Mozyrska, D., Girejko, E., Wyrwas, M.: Fractional nonlinear systems with sequential operators. Cent. Eur. J. Phys. 11, 1295–1303 (2013)

    Article  Google Scholar 

  36. Cheng, J.F.: The Theory of Fractional Difference Equations. Xiamen University Press, Xiamen (2011). (in Chinese)

    Google Scholar 

  37. Agarwal, R.P., El-Sayed, A.M.A., Salman, S.M.: Fractional-order Chua’s system: discretization, bifurcation and chaos. Adv. Differ. Equ. 320, (2013)

  38. Chen, F.L., Luo, X.N., Zhou, Y.: Existence results for nonlinear fractional difference equation. Adv. Differ. Equ. 713201 (2011)

  39. Holm, M.T.: The Theory of Discrete Fractional Calculus: Development and Application, PhD Thesis. University of Nebraska-Lincoln, Lincoln, Nebraska (2011)

  40. Tadjeran, C., Meerschaert, M.M., Scheffer, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  41. Scjerer, R., Kalla, S.L., Tang, Y.F., Huang, J.F.: The Grünwald-Letnikov method for fractional differential equations. Comput. Math. Appl. 62, 902–917 (2011)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (Grant No. 11301257), the Innovative Team Program of the Neijiang Normal University (Grant No. 13TD02), the Guangxi Natural Science Foundation (Grant No. 2013GXNSFBA019021), and the Scientific Research Foundation of GuangXi University (Grant No. XBZ120542).

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Authors and Affiliations

  1. Data Recovery Key Laboratory of Sichuan Province, Neijiang Normal University, Neijiang, 641100, Sichuan, China

    Guo-Cheng Wu & Sheng-Da Zeng

  2. Institute of Applied Nonlinear Science, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641100, Sichuan, China

    Guo-Cheng Wu

  3. Department of Mathematics and Computer Sciences, Cankaya University, 06530, Balgat, Ankara, Turkey

    Dumitru Baleanu

  4. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

  5. School of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Zhen-Guo Deng

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  1. Guo-Cheng Wu
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  2. Dumitru Baleanu
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  3. Sheng-Da Zeng
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  4. Zhen-Guo Deng
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Correspondence to Dumitru Baleanu.

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Wu, GC., Baleanu, D., Zeng, SD. et al. Discrete fractional diffusion equation. Nonlinear Dyn 80, 281–286 (2015). https://doi.org/10.1007/s11071-014-1867-2

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  • DOI: https://doi.org/10.1007/s11071-014-1867-2

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