Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media
Introduction
A great variety of diffusion problems in nature, which are referred to as normal diffusion, are satisfactorily described by the Fokker–Planck equation. However, in recent years the phenomenon of anomalous diffusion in disordered fractal media has attracted more and more attention [1]. O'shaughnessy and Procaccia [2] firstly derived approximate equation based on the properties of anomalous diffusion and obtained the exact solution of the equation. On the basis of theory of fractional calculus, Giona and Roman [3] restated diffusion equation in disordered fractal media and derived fractional differential equation form in disordered fractal media and developed a new study approach of diffusion kinetics in disordered fractal media. On the basis of the above mentioned studies, many literatures [4], [5], [6], [7], [8] proposed every kind of forms of anomalous diffusion equations in disordered fractal media from a different point of view and investigated their Cauchy problems. The Eq. (1) proposed by Metzler et al. [9] is the representative of themwhere is probability density. is Riemann–Liouville (R–L) fractional derivative of order defined aswhere and are the anomalous diffusion exponent and spectral dimension, respectively. The connection between them can be established by Hausdoff dimension via and is Gamma function.
In the present paper, we study fractional anomalous diffusion problem caused by an instantaneous point source in disordered fractal media. By applying the method of symmetry group of scaling transformations [10]; and the properties of -function, the analytical solutions of concentration distribution are given. We find that the solutions of classical diffusion problems are particular cases of this paper. At the same time, we derive the expressions of scattering function spectrum by using the formula of fractal Fourier transform. This physical quantity is of interest because it can be studied by X-ray and neutron scattering experiments on fractals. We also prove that the scattering function spectra still have the character of scaling function which stands for the feature of fractals.
Section snippets
Model and analytic solution of the problem
In order to solve the expression of concentration distribution caused by an instantaneous point source with strength M on the basis of Ref. [9] and after replacing ds by , we built fractional anomalous diffusion model as follows:where . Eqs. (3)–(5) are initial, boundary and conservation conditions, respectively. Using the Laplace transformation to (2)–(5), we
Scattering function of the problem
In general, physical measurements mostly reveal not but some spectral function, e.g. the Fourier–Laplace transformed spectral density , which can be studied by X-ray and neutron scattering experiments. The scattering function or dynamic structure factor defined in [3]: where . The fractal Fourier transform can be expressed by an ordinary Fourier sine transform as
Discussion of the results
In Fig. 1 we plotted concentration distribution curves for several sets of values of the anomalous diffusion exponent at fixed distance when . It can be seen that the greater the is, the more rapidly the concentration changes at initial time. In Fig. 2 we find that the concentration at the origin is increasing with the decrease of time for fixed values of and . In fact, we can discuss the singularities of concentration distribution at the origin in detail. For this
Acknowledgements
The authors express their gratitude to the referees of the paper for their fruitful advice and comments. This work was supported by the National Natural Science Foundation of China (Grant no.10272067) and by the Doctoral Foundation of the Education Ministry of P. R. China (Grant no. 20030422046).
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