Anomalous diffusion on the percolating networks

According to the standard diffusion equation, by introducing reasonably into a anomalous diffusion coefficient, the generalized diffusion equation, which describes anomalous diffusion on the percolating networks with a power-law distribution of waiting times, is derived in this paper. The solution of the generalized diffusion equation is obtained by using the method, which is used by Barta. The problems of anomalous diffusion on percolating networks with a power-law distribution of waiting times, which aren't solved by Barta, are resolved.     

Unique continuation property for anomalous slow diffusion equation

A Carleman estimate and the unique continuation of solutions for an anomalous diffusion equation with fractional time derivative of order 0 < α <1 are given. The estimate is derived through some subelliptic estimates for an operator associated to the anomalous diffusion equation using calculus of pseudo-differential operators.     

Space–time fractional diffusion on bounded domains

Fractional diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues. They are used in physics to model anomalous diffusion. This paper develops strong solutions of space–time fractional diffusion equations on bounded domains, as well as probabilistic representations of these solutions, which are useful for particle tracking codes.     

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis.     

Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion

The velocity field of generalized second order fluid with fractional anomalous diiusion caused by a plate moving impulsively in its own plane is investigated and the anomalous diffusion problems of the stress field and vortex sheet caused by this process are studied. Many previous and classical results can be considered as particular cases of this paper, such as the solutions of the fractional diffusion equations obtained by Wyss; the classical Rayleigh’s time-space similarity solution; the relationship between stress field and velocity field obtained by Bagley and co-worker and Podlubny’s results on the fractional motion equation of a plate. In addition, a lot of significant results also are obtained. For example, the necessary condition for causing the vortex sheet is that the time fractional diffusion index β must be greater than that of generalized second order fluid α; the establiihment of the vorticity distribution function depends on the time history of the velocity profile at a given point, and the time history can be described by the fractional calculus.     

Energy Characteristics of the Anomalous Diffusion Process

Theoretical and Mathematical Physics - We consider an anomalous diffusion model in which space-time nonlocalities are generated by singular zones forming sub- and superdiffusion transfer regimes....     

Anomalous diffusion approach to dielectric spectroscopy data with independent low- and high-frequency exponents

We advance a perspective outcome of tempered α-stable processes used in modeling of anomalous diffusion, a physical mechanism underlying the non-Debye relaxations. The tempered processes are characterized by a heavy tail truncation in time and have finite moments, but they also save some useful features of a purely skewed α-stable process. Due to these features, the relaxation phenomena get a transient character being shown in their asymptotic behavior. From the stochastic subordination scenario of the tempered anomalous diffusion we derive relaxation functions with independent low- and high-frequency exponents falling in the range (0, 1]. Those functions can be used to model all types of experimentally observed two-power-law relaxation patterns.     

A Diffusion Equation on Fractals in Random Media

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Finite element approximation for a modified anomalous subdiffusion equation

In this paper, we consider a modified anomalous subdiffusion equation (MASFE) for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. Firstly, a semi-discrete approximation for the MASFE is proposed. The stability and convergence of the semi-discrete approximation are discussed. Secondly, a finite element approximation for the MASFE is derived. The stability and convergence of the finite element approximation are investigated, respectively. Finally, some numerical examples are presented to demonstrate the effectiveness of theoretical analysis.     

Anomalous diffusion of particles with a finite free-motion velocity

The asymptotic behavior of the diffusion packet width is investigated in the anomalous diffusion of a particle with a finite free-motion velocity and a power distribution of free paths and waiting times in traps. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115 No. 1, pp. 154–160. April. 1998.     

Inverse problems of anomalous diffusion theory: An artificial neural network approach

The results are presented of computer simulation of the operation of a three-layer perceptron trained for solving inverse problems of anomalous diffusion theory. Several types of inverse problems are considered, including the problem of determining the Hurst exponent of a selfsimilar medium.     

Osgood type condition for the Volterra integral equations with bounded and nonincreasing kernels

This paper provides the necessary and sufficient Osgood type condition for the existence of blow-up solutions of Volterra equation with kernels being nonincreasing and bounded functions. Examples of such equations related to models of anomalous diffusion as well as some integral estimates of blow-up time are also presented.     

Analysis of diffusion process in fractured reservoirs using fractional derivative approach

The fractal geometry is used to model of a naturally fractured reservoir and the concept of fractional derivative is applied to the diffusion equation to incorporate the history of fluid flow in naturally fractured reservoirs. The resulting fractally fractional diffusion (FFD) equation is solved analytically in the Laplace space for three outer boundary conditions. The analytical solutions are used to analyze the response of a naturally fractured reservoir considering the anomalous behavior of oil production. Several synthetic examples are provided to illustrate the methodology proposed in this work and to explain the diffusion process in fractally fractured systems.     

Green’s function of time fractional diffusion equation and its applications in fractional quantum mechanics

Fractional equations, which have derivatives of noninteger order, are very successful in describing anomalous kinetics, transport, and chaos. To obtain the exact solutions of these equations, we need to find a more valid analytical method. In this paper, we propose a new technique for solving a class of time fractional partial differential equations. Then this method is successfully applied to find the Green’s functions; emphasis in the applications is given to the questions in anomalous diffusion and the time Schrödinger equation.     

Anomalous transport of particles in plasma physics

We investigate the long time/small mean-free-path asymptotic behavior of the solutions of a Vlasov–Lévy–Fokker–Planck equation and show that the asymptotic dynamics for the VLFP are described by an anomalous diffusion equation.     

Fractional Order Analysis of Sephadex Gel Structures: NMR Measurements Reflecting Anomalous Diffusion

We report the appearance of anomalous water diffusion in hydrophilic Sephadex gels observed using pulse field gradient (PFG) nuclear magnetic resonance (NMR). The NMR diffusion data was collected using a Varian 14.1 Tesla imaging system with a home-built RF saddle coil. A fractional order analysis of the data was used to characterize heterogeneity in the gels for the dynamics of water diffusion in this restricted environment. Several recent studies of anomalous diffusion have used the stretched exponential function to model the decay of the NMR signal, i.e., exp[-(bD)(α)], where D is the apparent diffusion constant, b is determined the experimental conditions (gradient pulse separation, durations and strength), and α is a measure of structural complexity. In this work, we consider a different case where the spatial Laplacian in the Bloch-Torrey equation is generalized to a fractional order model of diffusivity via a complexity parameter, β, a space constant, μ, and a diffusion coefficient, D. This treatment reverts to the classical result for the integer order case. The fractional order decay model was fit to the diffusion-weighted signal attenuation for a range of b-values (0 < b < 4,000 s-mm(-2)). Throughout this range of b values, the parameters β, μ and D, were found to correlate with the porosity and tortuosity of the gel structure.     

Nonfickian effect in time and space for diffusion processes

Diffusion processes are usually simulated using the classical diffusion equation. In certain scenarios, such equation induces anomalous behavior and consequently several improvements were introduced in the literature to overcome them. One of the most popular was the replacement of the diffusion equation by an integro‐differential equation. Such equation can be established considering a modification of Fick's mass flux where a delay in time is introduced. In this article, we consider mathematical models for diffusion processes that take into account a memory effect in time and space. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1589–1602, 2015     

Stochastic representation of solution to nonlocal-in-time diffusion

The aim of this paper is to derive a stochastic representation of the solution to a nonlocal-in-time evolution equation (with a historical initial condition), which serves a bridge between normal diffusion and anomalous diffusion. We first derive the Feynman–Kac formula by reformulating the original model into an auxiliary Caputo-type evolution equation with a specific forcing term subject to certain smoothness and compatibility conditions. After that, we confirm that the stochastic formula also provides the solution in the weak sense even though the problem data is nonsmooth. Finally, numerical experiments are presented to illustrate the theoretical results and the application of the stochastic formula.     

A spatial-fractional thermal transport model for nanofluid in porous media

In this paper, a spatial fractional-order thermal transport equation with the Caputo derivative is proposed to describe convective heat transfer of nanofluids within disordered porous media in boundary layer flow. This equation arises naturally when the effect of anomalous migration of nanoparticles on heat transfer is considered. The numerical results show that local Nusselt numbers of four different kinds of nanofluids are all inversely proportional to the fractional derivative exponent β. Based on this finding, it is concluded that the anomalous diffusion of nanoparticles improves the convective heat transfer of nanofluids and the space fractional thermal transport equation may serve as a candidate model for studying nanofluids. Additionally, the effects of other involved physical parameters on temperature distribution and Nusselt number are presented and analyzed.     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.