Analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media

https://doi.org/10.1016/j.ijnonlinmec.2004.07.023Get rights and content

Abstract

A theoretical analysis of fractional anomalous diffusion caused by an instantaneous point source in disordered fractal media is studied. Using the method of symmetry group of scaling transformations and the H-function, the analytical solutions of concentration distribution are given. At the same time we derive the expressions of scattering function spectrum.The result shows that the scattering function spectra still have the properties of scaling function. The scattering functions of point source, line source and area source in regular Euclidean space can be regarded as particular cases of this paper and are included in this paper. At the end of the paper we discuss the asymptotic behaviors of the solution in detail. The results of this paper can be taken to be the fundamental solutions for every kind of boundary value problems of fractional anomalous diffusion 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 themDt2/dw0p(r,t)=r1-dsrrds-1rp(r,t),where p(r,t) is probability density. Dt2/dw0 is Riemann–Liouville (R–L) fractional derivative of order 2/dw defined asDt2/dw0p(r,t)=Γ-1(1-2/dw)t0t(t-τ)-2/dwp(r,τ)dτ,where dw and ds are the anomalous diffusion exponent and spectral dimension, respectively. The connection between them can be established by Hausdoff dimension df via 2df=dsdw and Γ(·) 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 H-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 2df/dw, we built fractional anomalous diffusion model as follows:Dt2/dw0c(r,t)=r1-(2df/dw)rr(2df/dw)-1rc(r,t),c(r,0)=0,r>0,limrc(r,t)=0,t>0,0ωdfrdf-1c(r,t)dr=M=const,t0,where ωdf=2πdf/2/Γ(df/2). 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 c(r,t) but some spectral function, e.g. the Fourier–Laplace transformed spectral density c˜(k,s), which can be studied by X-ray and neutron scattering experiments. The scattering function or dynamic structure factor defined in [3]: c˜(k,ω)=Rec˜(k,s)=Rec˜(k,iω),where c˜(k,s)=exp(ik·r)dr0exp(-st)c(r,t)dt. The fractal Fourier transform Ff{f(r);k} can be expressed by an ordinary Fourier sine transform Fs{rdf-2f(r);k} asFf{f(r);k}=12πN/22πdf/2k-1

Discussion of the results

In Fig. 1 we plotted concentration distribution curves for several sets of values of the anomalous diffusion exponent dw at fixed distance r=1 when df=1.8. It can be seen that the greater the dw is, the more rapidly the concentration changes at initial time. In Fig. 2 we find that the concentration at the origin c(0,t) is increasing with the decrease of time for fixed values of df and dw. 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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