Scaling, Self-Similarity, and Systems of Fractional OrderView this Special Issue
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Jun-Sheng Duan, Ai-Ping Guo, Wen-Zai Yun, "Similarity Solution for Fractional Diffusion Equation", Abstract and Applied Analysis, vol. 2014, Article ID 548126, 5 pages, 2014. https://doi.org/10.1155/2014/548126
Similarity Solution for Fractional Diffusion Equation
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 , which is just the fundamental solution of the fractional diffusion equation, was expressed in terms of Fox functions.
Standard diffusion in -dimensional space, where is a positive integer, is a process described by Gaussian distribution. A main feature of the process is the linear relation between the mean square displacement and time; namely, . Some anomalous diffusion phenomena that take place in impure media, biological tissues, and porous media can be simulated by the diffusion model in fractals [1–6]. In recent years, the fractal theory has been developed rapidly, and it was found to be closely related to the anomalous diffusion phenomena [3–12].
In fractal media, the geometric obstacles existing on all length scales slow down the particle motion in a random walk. The mean square displacement behaves as  where is the anomalous diffusion exponent. The numerical simulation found that on a large class of fractal structures the general form of the probability density function that the walker is at distance at time from its starting point at time obeys asymptotically a non-Gaussian shape of the form [2, 3] where and is the fractal Hausdorff dimension.
In order to simulate the diffusion phenomena in fractal media, some scholars have introduced fractional diffusion equations [4, 5, 11–13]. In this paper, we consider the fractional diffusion equation [5, 13]: where , is the spectral dimension of the fractal, and the fractional time derivative on the left hand side of (3) is defined as the convolution integral [14–20]: where is Euler's gamma function. In the limit case, and , (3) reduces to the standard -dimensional diffusion equation.
The fractional calculus has been applied to many fields in science and engineering, such as viscoelasticity, anomalous diffusion, biology, chemistry, and control theory [5, 11–13, 15, 19–22]. Researches on the fractional differential equations attract much attention [15, 23–28]. For linear fractional differential equations, the integral transforms, including the Laplace, Fourier, and Mellin transforms, are usually used to obtain analytic solutions.
We note that the probability density function is just the fundamental solution of the fractional diffusion equation. The similarity method was used by Gorenflo et al. , Wyss , and Buckwar and Luchko  for solving problems of time fractional partial differential equations in one-dimensional case.
2. Derivation of Similarity Solution
First we determine a symmetric group of scaling transformations where is a parameter and , are constants to be determined. Applying the group of scaling transformations (7), the fractional derivative is converted as follows: where . Hence the problem (3)–(5) is invariant under the group (7) if and only if So the symmetric group of scaling transformations is determined:
Eliminating the parameter leads to two invariants: We denote the two invariants of the group of the scaling transformation as
Next we use the transformation to determine the equations for the similarity solution of the problem (3)–(5). Calculating derivative we have where For the left hand side of (3), we introduce the new integral variable we obtain , and Letting we rewrite (18) as
Applying Mellin transform with respect to to both sides of (21), we get Calculating integrations we obtain Mellin transform of the function : Inserting (25) into (24) and then replacing by we obtain the difference equation for the function :
In order to solve the difference equation, we introduce and , and rewrite (26) into A particular solution of (27) is where is an arbitrary constant. For the solution of (27), we can multiply by any function which satisfies .
We notice that is a Mellin transform defined only in some strip from the conditions (22). So (26) is valid only in the overlap of the two strips and , and there is no such overlap unless . Thus cannot have poles; otherwise, it would have a row of poles separated exactly by one unit. In addition, cannot grow faster than as in the inversion strip; otherwise the inversion integral would diverge. Thus is a bounded entire function and equals a constant by Liouville's theorem.
Therefore, has only the form of (28) and we have
It follows from (22) that . Thus we have
The inverse Mellin transform of (29) is Replacing by and using the definition of Fox functions we obtain [34, 35] Inserting the expressions into (13) and using properties of Fox functions, we obtain the probability density function in terms of the Fox function:
For a large class of fractal structures, the spectral dimension  satisfies ; that is, . So the Fox function in (33) can be expanded into a series by using residue theorem on the simple poles: The series representation for the probability density is calculated to be
3. Discussions and Conclusions
In Figures 1 and 2, we plot the curves of versus and versus , respectively, for and different values of . In Figures 3 and 4, we plot the curves of versus and versus , respectively, for and different values of . The figures display that, as the anomalous diffusion exponent increases, the peak value of the probability density function at decreases. In addition, as the fractal Hausdorff dimension increases from 1 to 1.5, the peak value of at decreases.
Compared with the similarity method for classic partial differential equations, the similarity method for fractional diffusion equation involves the similarity integral variable , and the reduction equation is an integro-ordinary differential equation for the similarity solution. The obtained probability density is just the fundamental solution of the fractional diffusion equation.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (11201308) and the Innovation Program of Shanghai Municipal Education Commission (14ZZ161).
- 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.
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