Mass transfer in nonuniform media

The lattice model was used to derive equations describing diffusion in a nonuniform medium in the absence of local equilibrium at nonzero temperature gradients. Equations are obtained that extend a diffusion equation with fractional derivative in time and a nonlinear diffusion equation, which was previously obtained within the framework of a generalized thermodynamic approach, to cover diffusion in a medium with a nonuniform temperature field.     

反常扩散与分数阶对流-扩散方程

反常扩散现象在自然界和社会系统中广泛存在.考虑了扩散过程的时间相关和时空相关性，用非局域性的处理方法，在传统的二阶对流 扩散方程基础上，得到了分数阶对流 扩散方程，以此方程来描述反常扩散.在此方程中，弥散项和对时间的导数为分数阶导数所代替.由此分数阶对流 扩散方程，对传统的费克扩散定律进行推广，得到了广义的分数费克扩散定律，分数费克扩散定律说明某时刻空间中某点的流量不仅与其领域内的浓度梯度有关，而且与整个空间中其他不同点的粒子浓度、浓度变化的历史，甚至初始时刻的浓度有关.讨论了方程的解——分数稳定分布，并由此说明了扩散运动的平均平方位移是运移时间的非线性函数. 关键词： 扩散 分数阶微积分 稳定分布（Lévy分布） 费克扩散定律     

Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative

We examine a numerical method to approximate to a fractional diffusion equation with the Riesz fractional derivative in a finite domain, which has second order accuracy in time and space level. In order to approximate the Riesz fractional derivative, we use the “fractional centered derivative” approach. We determine the error of the Riesz fractional derivative to the fractional centered difference. We apply the Crank–Nicolson method to a fractional diffusion equation which has the Riesz fractional derivative, and obtain that the method is unconditionally stable and convergent. Numerical results are given to demonstrate the accuracy of the Crank–Nicolson method for the fractional diffusion equation with using fractional centered difference approach.     

Results for a fractional diffusion equation with a nonlocal term in spherical symmetry

We obtain time dependent solutions for a fractional diffusion equation containing a nonlocal term by considering the spherical symmetry and using the Green function approach. The nonlocal term incorporated in the diffusion equation may also be related to the spatial and time fractional derivative and introduces different regimes of spreading of the solution with the time evolution. In addition, a rich class of anomalous diffusion processes may be described from the results obtained here.     

Numerical algorithm for the time fractional Fokker–Planck equation

Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^min{1+2α,2})+O(h²), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for α=1.0 with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for α=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.     

The time fractional diffusion-wave equation

The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0<β≤1/2 or 1/2<β≤1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M (z; β), where z is the similarity variable. Such function, which reduces to the well-known Gaussian function for β=1/2 (ordinary diffusion), is proved to be an entire function of Wright type.     

Exact Solutions for Fractional Differential-Difference Equations by (G′/G)-Expansion Method with Modified Riemann-Liouville Derivative

In this paper, the ($G′/G$)-expansion method is suggested to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann–Liouville derivative. The fractional complex transform is proposed to convert a fractional partial differential difference equation into its differential difference equation of integer order. With the aid of symbolic computation, we choose nonlinear lattice equations to illustrate the validity and advantages of the algorithm. It is shown that the proposed algorithm is effective and can be used for many other nonlinear lattice equations in mathematical physics and applied mathematics.     

间歇湍流的分数阶动力学

间歇湍流意味着湍流涡旋并不充满空间,其维数介于2和3之间.湍流扩散为超扩散,且概率密度分布具有长尾特征.本文将流体力学的Navier-Stokes(NS)方程中的黏性项用分数阶的拉普拉斯算子表达.分析表明,分数阶拉普拉斯的阶数α和间歇湍流的维数D相联系.对于均匀各向同性的Kolmogorov湍流α=2,即用整数阶NS方程描述.而对于间歇性湍流,一定用分数阶的NS方程来描述.对于Kolmogorov湍流,扩散方差正比于t3,即Richardson扩散.而对于间歇性湍流,扩散方差要比Richardson扩散更强.     

时间分数阶Boussinesq方程的李对称分析

本文利用李群分析方法研究了时间分数阶Boussinesq方程,得到了该方程的李点对称,并把该方程约化为Erdelyi-Kobe分数阶常微分方程. 本文的行文过程也说明了李群分析方法对于约化分数阶非线性发展方程是有效的. 关键词： 李对称分析方法 时间分数阶Boussinesq方程 广义Riemann-Liouville导数 Erdelyi-Kober微分算子     

Continuum description of anomalous diffusion on a comb structure

Anomalous diffusion on a comb structure consisting of a one-dimensional backbone and lateral branches (teeth) of random length is considered. A well-defined classification of the trajectories of random walks reduces the original problem to an analysis of classical diffusion on the backbone, where, however, the time of this process is a random quantity. Its distribution is dictated by the properties of the random walks of the diffusing particles on the teeth. The feasibility of applying mean-field theory in such a model is demonstrated, and the equation for the Green’s function with a partial derivative of fractional order is obtained. The characteristic features of the propagation of particles on a comb structure are analyzed. We obtain a model of an effective homogeneous medium in which diffusion is described by an equation with a fractional derivative with respect to time and an initial condition that is an integral of fractional order. Zh. éksp. Teor. Fiz. 114, 1284–1312 (October 1998)     

Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation

Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.     

Fractional generalized Langevin equation approach to single-file diffusion

Fractional generalized Langevin equation with external force is used to model single-file diffusion. It is found that for external force that varies with power law the solution for such a fractional Langevin equation gives the correct short and long time behavior for the mean square displacement of single-file diffusion when appropriate choice of parameters associated with fractional generalized Langevin equation are used. By considering some special cases of the fractional generalized Langevin equation, a new class of closed analytic expressions for the mean square displacement of single-file diffusion can be obtained. The effective Fokker-Planck equation associated with single-file diffusion is briefly considered.     

Model of superdiffusion

A locally nonequilibrium model of superdiffusion is proposed that is based on the partition of the set of diffusing particles into groups according to the flight length of these particles. The process of diffusion is described in terms of partial concentrations of particles belonging to different groups. As special limit cases, the model yields equations with fractional time derivative and the so-called porous medium equation. The basic equations of the model are Markov equations; therefore, they easily include reaction terms. The model can be applied to describing the types of diffusion in which the diffusing particles are in free flight most of the time.     

Exact Solutions for Fractional Differential-Difference Equations by an Extended Riccati Sub-ODE Method

In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann-Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established.     

The influence of fractality on the time evolution of the diffusion process

In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics. On the other hand, the physical origins of the order of derivative namely α in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters α and q are enlightened in connection with fractality of space and the memory effect. It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method.     

Non-steady-state electrical conduction of polymers in the model with fractional integro-differentiation

A simple dynamic model with a fractional time derivative is considered for conducting polymers. The elementary theory of electrical conduction is developed using the fractional equation of motion. In the framework of the model under consideration, the relaxation of the velocity of charge carriers is described by the Mittag-Leffler function. A kinetic equation with the fractional derivative is derived. The electrical conductivity is calculated to a first approximation with the use of the derived kinetic equation. It is demonstrated that the spectral characteristic of non-steady-state current fluctuations in a polymer should be proportional to 1/f.     

Analytical approach for the fractional differential equations by using the extended tanh method

In this study, we consider analytical solutions of space–time fractional derivative foam drainage equation, the nonlinear Korteweg–de Vries equation with time and space-fractional derivatives and time-fractional reaction–diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann–Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.     

Residence Time Statistics for Normal and Fractional Diffusion in a Force Field

We investigate statistics of occupation times for an over-damped Brownian particle in an external force field, using a backward Fokker–Planck equation introduced by Majumdar and Comtet. For an arbitrary potential field the distribution of occupation times is expressed in terms of solutions of the corresponding first passage time problem. This general relationship between occupation times and first passage times, is valid for normal Markovian diffusion and for non-Markovian sub-diffusion, the latter modeled using the fractional Fokker–Planck equation. For binding potential fields we find in the long time limit ergodic behavior for normal diffusion, while for the fractional framework weak ergodicity breaking is found, in agreement with previous results of Bel and Barkai on the continuous time random walk on a lattice. For non-binding cases, rich physical behaviors are obtained, and classification of occupation time statistics is made possible according to whether or not the underlying random walk is recurrent and the averaged first return time to the origin is finite. Our work establishes a link between fractional calculus and ergodicity breaking.     

Anomalous diffusion and charge relaxation on comb model: exact solutions

The random walks on the comb structure are considered. It is shown that due to fingers a diffusion has an anomalous character, that is an r.m.s. displacement depends on time by a power way with exponent . The generalized diffusion equation for an anomalous case is deduced. It essentially differs from a usual diffusion equation in the continuity equation form: instead of the first time derivative, the time derivative of fractal order appears. In the second part the charge relaxation on the comb structure is studied. A non-Maxwell character is established. The reason is that the electric field has three components, but a charge may relax only along some conducting lines.