Abstract and Applied Analysis

Similarity Solution for Fractional Diffusion Equation

Jun-Sheng Duan, Ai-Ping Guo, and Wen-Zai Yun

Full-text: Open access

Abstract

Fractional diffusion equation in fractal media is an integropartial differential equation parametrized by fractal Hausdorff dimension and anomalous diffusion exponent. In this paper, the similarity solution of the fractional diffusion equation was considered. Through the invariants of the group of scaling transformations we derived the integro-ordinary differential equation for the similarity variable. Then by virtue of Mellin transform, the probability density function p(r,t), which is just the fundamental solution of the fractional diffusion equation, was expressed in terms of Fox functions.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 548126, 5 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1425047789

Digital Object Identifier
doi:10.1155/2014/548126

Mathematical Reviews number (MathSciNet)
MR3182289

Zentralblatt MATH identifier
07022603

Citation

Duan, Jun-Sheng; Guo, Ai-Ping; Yun, Wen-Zai. Similarity Solution for Fractional Diffusion Equation. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 548126, 5 pages. doi:10.1155/2014/548126. https://projecteuclid.org/euclid.aaa/1425047789


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References

  • B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York, NY, USA, 1982.
  • S. Havlin and D. Ben-Avraham, “Diffusion in disordered media,” Advances in Physics, vol. 51, no. 1, pp. 187–292, 2002.
  • D. Liu, H. Li, F. Chang, and L. Lin, “Anomalous diffusion on the percolating networks,” Fractals, vol. 6, no. 2, pp. 139–144, 1998.
  • F.-Y. Ren, J.-R. Liang, and X.-T. Wang, “The determination of the diffusion kernel on fractals and fractional diffusion equation for transport phenomena in random media,” Physics Letters A, vol. 252, no. 3-4, pp. 141–150, 1999.
  • Q. Zeng and H. Li, “Diffusion equation for disordered fractal media,” Fractals, vol. 8, no. 1, pp. 117–121, 2000.
  • C. Cattani, “Fractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010, Article ID 507056, 31 pages, 2010.
  • M. Li and W. Zhao, “On bandlimitedness and lag-limitedness of fractional Gaussian noise,” Physica A, vol. 392, no. 9, pp. 1955–1961, 2013.
  • M. Li, “A class of negatively fractal dimensional Gaussian random functions,” Mathematical Problems in Engineering, vol. 2011, Article ID 291028, 18 pages, 2011.
  • M. Li, C. Cattani, and S.-Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011.
  • C. Cattani and G. Pierro, “On the fractal geometry of DNA by the binary image analysis,” Bulletin of Mathematical Biology, vol. 75, no. 9, pp. 1544–1570, 2013.
  • R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.
  • M. Giona and H. E. Roman, “Fractional diffusion equation for transport phenomena in random media,” Physica A, vol. 185, no. 1–4, pp. 87–97, 1992.
  • R. Metzler, W. G. Glöckle, and T. F. Nonnenmacher, “Fractional model equation for anomalous diffusion,” Physica A, vol. 211, no. 1, pp. 13–24, 1994.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, Amsterdam, The Netherlands, 1993.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
  • K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.
  • D. Băleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  • F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010.
  • M. Y. Xu and W. C. Tan, “Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion,” Science in China A, vol. 44, no. 11, pp. 1387–1399, 2001.
  • C. P. Li, W. H. Deng, and D. Xu, “Chaos synchronization of the Chua system with a fractional order,” Physica A, vol. 360, no. 2, pp. 171–185, 2006.
  • J.-S. Duan, “Time- and space-fractional partial differential equations,” Journal of Mathematical Physics, vol. 46, no. 1, Article ID 013504, pp. 13504–13511, 2005.
  • J.-S. Duan, “The periodic solution of fractional oscillation equation with periodic input,” Advances in Mathematical Physics, vol. 2013, Article ID 869484, 6 pages, 2013.
  • J. S. Duan, R. Rach, D. Baleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applications to fractional differential equations,” Communications in Fractional Calculus, vol. 3, no. 2, pp. 73–99, 2012.
  • F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, “Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 12–20, 2007.
  • Z. H. Wang and X. Wang, “General solution of the Bagley-Torvik equation with fractional-order derivative,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 5, pp. 1279–1285, 2010.
  • A. M. Yang, C. Cattani, H. Jafari, and X. J. Yang, “Analytical solutions of the onedimensional heat equations arising in fractal transient conduction with local fractional derivative,” Abstract and Applied Analysis, vol. 2013, Article ID 462535, 5 pages, 2013.
  • G. W. Bluman and S. C. Anco, Symmetry and Integration Methods for Differential Equations, Springer, New York, NY, USA, 2002.
  • R. Gorenflo, Y. Luchko, and F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 175–191, 2000.
  • W. Wyss, “The fractional diffusion equation,” Journal of Mathematical Physics, vol. 27, no. 11, pp. 2782–2785, 1986.
  • E. Buckwar and Y. Luchko, “Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 81–97, 1998.
  • B. Davies, Integral Transforms and Their Applications, Springer, New York, NY, USA, 3rd edition, 2002.
  • A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines, John Wiley & Sons, New Delhi, India, 1978.
  • H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian, New Delhi, India, 1982. \endinput