Tempered stable Lévy motion and transient super-diffusion

The space-fractional diffusion equation models anomalous super-diffusion. Its solutions are transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool these jumps. A tempered fractional diffusion equation governs the transition densities, which progress from super-diffusive early-time to diffusive late-time behavior. This article provides finite difference and particle tracking methods for solving the tempered fractional diffusion equation with drift. A temporal and spatial second-order Crank-Nicolson method is developed, based on a finite difference formula for tempered fractional derivatives. A new exponential rejection method for simulating tempered Lévy stables is presented to facilitate particle tracking codes.     

Modeling non-Darcian flow and solute transport in porous media with the Caputo–Fabrizio derivative

In this study, the non-Darcian flow and solute transport in porous media are modeled with a revised Caputo derivative called the Caputo–Fabrizio fractional derivative. The fractional Swartzendruber model is proposed for the non-Darcian flow in porous media. Furthermore, the normal diffusion equation is converted into a fractional diffusion equation in order to describe the diffusive transport in porous media. The proposed Caputo–Fabrizio fractional derivative models are addressed analytically by applying the Laplace transform method. Sensitivity analyses were performed for the proposed models according to the fractional derivative order. The fractional Swartzendruber model was validated based on experimental data for water flows in soil–rock mixtures. In addition , the fractional diffusion model was illustrated by fitting experimental data obtained for fluid flows and chloride transport in porous media. Both of the proposed fractional derivative models were highly consistent with the experimental results.     

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order 0 < α < 1 ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent withO(Τ +h ²), where Τ andh are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A new nonlocal nonlinear diffusion of image processing

A novel nonlocal nonlinear diffusion is analyzed which has proven useful as a denoising tool in image processing. The equation can be viewed as a new paradigm for the regularization of the well-known Perona-Malik equation. The regularization is implemented via nonlinearity intensity reduction through fractional derivatives. Well-posedness in the weak setting is established. Global existence and convergence to the average holds in the purely diffusive limit whereas an interesting dynamic behavior is engendered by the presence of nontrivial equilibria as the intensity of the nonlinearity is increased and comes close to the one of Perona-Malik.     

Solutions for a fractional diffusion equation with noninteger dimensions

We investigate a fractional diffusion equation with a nonlocal reaction term by using the Green function approach. We also consider a modified spatial operator in order to cover situations characterized by a noninteger dimension. The results show a nonusual spreading of the initial condition which can be connected to a rich class of anomalous diffusive processes.     

The role of the Fox–Wright functions in fractional sub-diffusion of distributed order

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin–Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox–Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.     

Global regularity of generalized magnetic Benard problem

We study the magnetic Bénard problem in two‐dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.     

Continuous pseudospectral methods for the neutron diffusion equation in 1D geometries

The P_L equations are classical approximations to the neutron transport equation that admit a diffusive form. The diffusive form of the P₁ approximation is known as the neutron diffusion equation. Different methods based on the expansion of the neutron flux in terms of a continuous basis of polynomials have been developed for the neutron diffusion equation and tested using two 1D benchmark problems.     

Inverse problem of fractional calculus of variations for partial differential equations

The current paper aims at finding out a Lagrangian structure for some partial differential equations including the Stokes equations, the fractional wave equation, the diffusion or fractional diffusion equations, using the fractional embedding theory of continuous Lagrangian systems.     

The development of a bedform disturbance in an alluvial river or channel

The evolution of a localized disturbance imposed upon an otherwise uniform alluvial flow is considered. For small disturbances a linearized theory is developed which shows that the initial disturbance splits into two modes. One mode is stationary and purely diffusive while the other mode propagates. The propagating mode may exhibit diffusion or, for sufficiently high Froude numbers instability of the roll-wave type. The theory provides the relevant diffusion, propagation and instability time scales associates with the two modes.For finite amplitude disturbances, a weakly nonlinear theory is considered. Again the disturbance separates into two modes. The stationary mode remains as a solution of the diffusion equation, but the propagating mode is now governed by a Burger's equation.     

Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.     

Quenching phenomenon of a time‐fractional diffusion equation with singular source term

In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

Numerical approach to chaotic pattern formation in diffusive predator–prey system with Caputo fractional operator

This paper is primarily concern with the formulation and analysis of a reliable numerical method based on the novel alternating direction implicit finite difference scheme for the solution of the fractional reaction–diffusion system. In the work, the integer first‐order derivative in time is replaced with the Caputo fractional derivative operator. As a case study, the dynamics of predator–prey model is considered. In order to provide a good guidelines on the correct choice of parameters for the numerical simulation of full fractional reaction–diffusion system, its linear stability analysis is also examined. The resulting scheme is applied to solve both self‐diffusion and cross‐diffusion problems in two‐dimensions. We observed in the experimental results a range of spatiotemporal and chaotic structures that are related to Turing pattern. It was also discovered in the simulations that cross‐diffusive case gives rise to spatial patterns faster than the diffusive case. Apart from chaotic spiral‐like structures obtained in this work, it should also be mentioned that Turing patterns such as stationary spots and stripes are obtainable, depending on the initial and parameters choices.     

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.     

A possible way of estimating options with stable distributed underlying asset prices

Option pricing theory is considered when the underlying asset price satisfies a stochastic differential equation which is driven by random motions generated by stable distributions. The properties of the stable distributions are discussed and their connection with the theory of fractional Brownian motion is noted. This approach attempts to generalize the classical Black–Scholes formulation, to allow for the presence of fat tails in the distribution of log prices which leads to a diffusion equation involving fractional Brownian motion. The resulting option pricing via a hedging strategy approach is independently derived by constructing a backward Kolmogorov equation for a simple trinomial model where the probabilities are assumed to satisfy a particular fractional Taylor series due to Dzherbashyan and Nersesyan. To effect this development, some knowledge of fractional integration and differentiation is required so this is briefly reviewed. Consideration is also given to a different hedging strategy approach leading to a fractional Black–Scholes equation involving the market price of risk. Modification to the model is also considered such as the impact of transaction costs. A simple example of American options is also considered.     

Thermoelasticity that uses fractional heat conduction equation

A survey of nonlocal generalizations of the Fourier law and heat conduction equation is presented. More attention is focused on the heat conduction with time and space fractional derivatives and on the theory of thermal stresses based on this equation.     

Traveling wavefronts in temporally discrete reaction–diffusion equations with delay

This paper is concerned with the existence of traveling wavefronts of a temporally discrete reaction–diffusion equation with delay. By using monotone iteration and upper–lower solution technique, the existence of traveling wavefronts for the temporally discrete reaction–diffusion equation with delay is established. As an application, we consider an abstract diffusive equation, which includes a single species diffusive model as a particular case. Our result implies the temporally discrete model is a good approximation of corresponding continuous time model in sense of propagation.     

The Averaging Principle for Stochastic Fractional Partial Differential Equations with Fractional Noises

The purpose of this paper is to establish an averaging principle for stochastic fractional partial differential equation of order α > 1 driven by a fractional noise. We prove the existence and uniqueness of the global mild solution for the considered equation by the fixed point principle. The solutions for SPDEs with fractional noises can be approximated by the solution for the averaged stochastic systems in the sense of p-moment under some suitable assumptions.     

分数阶微分方程边值问题研究简介

分数阶微积分是一个古老而又新颖的课题,近30年来,由于在包括分形现象在内的物理、工程等诸多应用学科领域应用的拓展,激发了科研人员对分数阶微积分的巨大热情。分数阶微分方程现在已应用于分数物理学、混沌与湍流、粘弹性力学与非牛顿流体力学、高分子材料的解链、自动控制理论、化学物理、随机过程和反常扩散等许多科学领域。分数阶微分方程边值问题是非线性常微分方程理论研究中一个活跃而成果丰硕的领域。本文讨论了分数阶微分方程边值问题的一些理论,介绍了作者的著作《分数阶微分方程边值问题理论及应用》的基本内容。