Scaling, Self-Similarity, and Systems of Fractional OrderView this Special Issue
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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).
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