Space Fractional Derivatives Research Articles

Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation

In this paper, we consider a space–time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α ∈ (0, 1], and the first-order and second-order space derivatives by the Riemman–Liouville fractional derivatives of order β ∈ (0, 1] and of order γ ∈ (1, 2], respectively. For the space fractional derivatives D x β u ( x , t ) and D x γ u ( x , t ) , we adopted the Grünwald formula and the shift Grünwald formula, respectively. We propose an implicit difference method (IDM) and an explicit difference method (EDM) to solve this equation. Stability and convergence of these methods are discussed. Using mathematical induction, we prove that the IDM is unconditionally stable and convergent, but the EDM is conditionally stable and convergent. Numerical results are in good agreement with theoretical analysis.

Generalized fractional Schrödinger equation with space-time fractional derivatives

In this paper the generalized fractional Schrödinger equation with space and time fractional derivatives is constructed. The equation is solved for free particle and for a square potential well by the method of integral transforms, Fourier transform and Laplace transform, and the solution can be expressed in terms of Mittag-Leffler function. The Green function for free particle is also presented in this paper. Finally, we discuss the relationship between the cases of the generalized fractional Schrödinger equation and the ones in standard quantum.

Fractional diffusion equation with an absorbent term and a linear external force: Exact solution

We investigate the solutions of a generalized diffusion equation which contains space and time fractional derivatives by taking an absorbent (or source) term and a linear external force into account. They are expressed in terms of the Fox H functions and, for some special cases, they are directly related to the Levy distributions. We also analyze the first passage time distribution for the case characterized by the absence of absorbent (or source) term and external force for a semi-infinite interval with absorbent boundary conditions. We also discuss the connection of the results presented here with the anomalous diffusion.

Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method

Abstract Based on the symbolic computation system Maple , Adomian decomposition method, developed for differential equations of integer order, is directly extended to derive explicit and numerical solutions of the fractional KdV–Burgers equation. The fractional derivatives are described in the Caputo sense. According to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional KdV–Burgers equation with time- and space-fractional derivatives. Finally, the solutions of our model equation are calculated in the form of convergent series with easily computable components.

Pattern formation in a fractional reaction–diffusion system

We investigate pattern formation in a fractional reaction–diffusion system. By the method of computer simulation of the model of excitable media with cubic nonlinearity we are able to show structure formation in the system with time and space fractional derivatives. We further compare the patterns obtained by computer simulation with those obtained by simulation of the similar system without fractional derivatives. As a result, we are able to show that nonlinearity plays the main role in structure formation and fractional derivative terms change the transient dynamics. So, when the order of time derivative increases and approaches the value of 1.5, the special structure formation switches to homogeneous oscillations. In the case of space fractional derivatives, the decrease of the order of these derivatives leads to more contrast dissipative structures. The variational principle is used to find the approximate solution of such fractional reaction–diffusion model. In addition, we provide a detailed analysis of the characteristic dissipative structures in the system under consideration.

Non-perturbative analytical solutions of the space- and time-fractional Burgers equations

Abstract Non-perturbative analytical solutions for the generalized Burgers equation with time- and space-fractional derivatives of order α and β, 0

Error analysis of an explicit finite difference approximation for the space fractional diffusion equation with insulated ends

The space fractional diffusion equation (?) is obtained from the classical diffusion equation by replacing the second space derivative by a fractional derivative of order ?, 1 ? ? ? 2 . Numerical methods associated with integer-order differential equation, have been extensively treated. On the other hand, studies of the numerical methods and error estimates of fractional order differential equations are quite limited to date. Here, we propose an explicit finite difference approximation (?) for ?. An error analysis of the explicit numerical method for ? with insulated ends is discussed. We derive the scaling restriction of the stability and convergence of the explicit numerical method. Finally, some numerical results show the diffusion behaviour according to the order of space-fractional derivative and demonstrate that our ? is computationally simple for ?.

Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives

This paper is concerned with establishing necessary or sufficient conditions for the existence of solutions to evolution equations with fractional derivatives in space and time. The Fujita exponent is determined. Then, these results are extended to systems of reaction–diffusion equations. Our new results shed lights on important practical questions.

Space-time fractional derivative operators

Evolution equations for anomalous diffusion employ fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. This paper develops the mathematical foundations of those operators.

Nondiffusive Transport in Plasma Turbulence: A Fractional Diffusion Approach

Numerical evidence of nondiffusive transport in three-dimensional, resistive pressure-gradient-driven plasma turbulence is presented. It is shown that the probability density function (pdf) of tracer particles' radial displacements is strongly non-Gaussian and exhibits algebraic decaying tails. To model these results we propose a macroscopic transport model for the pdf based on the use of fractional derivatives in space and time that incorporate in a unified way space-time nonlocality (non-Fickian transport), non-Gaussianity, and nondiffusive scaling. The fractional diffusion model reproduces the shape and space-time scaling of the non-Gaussian pdf of turbulent transport calculations. The model also reproduces the observed superdiffusive scaling.

Fractional diffusion in plasma turbulence

Transport of tracer particles is studied in a model of three-dimensional, resistive, pressure-gradient-driven plasma turbulence. It is shown that in this system transport is anomalous and cannot be described in the context of the standard diffusion paradigm. In particular, the probability density function (pdf) of the radial displacements of tracers is strongly non-Gaussian with algebraic decaying tails, and the moments of the tracer displacements exhibit superdiffusive scaling. To model these results we present a transport model with fractional derivatives in space and time. The model incorporates in a unified way nonlocal effects in space (i.e., non-Fickian transport), memory effects (i.e., non-Markovian transport), and non-Gaussian scaling. There is quantitative agreement between the turbulence transport calculations and the fractional diffusion model. In particular, the model reproduces the shape and space-time scaling of the pdf, and the superdiffusive scaling of moments.

Fractal scaling of fractional diffusion processes

The fractal growth of fractional diffusion is analyzed from the viewpoint of the influence of fractional derivative order on scaling exponents. Fractional diffusion is considered here as deterministic with stochastic forcing, and with time and space fractional derivatives defined as the Caputo and Riesz–Feller forms, respectively. Fractal growth, as characterized by a dynamic scaling exponent, z, roughness, α, and growth exponent, β, is model-dependent, and we show that the exponents are distinguished by the order of fractional derivatives and the form of stochastic process. Finally, scaling exponents for nonconservative (or uncorrelated) and correlated stochastic processes are found in the lowest-order linear fractional differential equations, and the prospect for construction of stochastic nonlinear fractional diffusion evolution equations is explored here. In application to signal processing, this amounts to modeling dynamic processes (signal evolution, image growth, surface and interface growth, structural growth, etc.) and their equivalence (or approximation) in fractional differencing models of discrete data.

Crossover in diffusion equation: anomalous and normal behaviors.

Ubiquitous phenomena exist in nature where, as time goes on, a crossover is observed between different diffusion regimes (e.g., anomalous diffusion at early times which becomes normal diffusion at long times, or the other way around). In order to focus on such situations we have analyzed particular relevant cases of the generalized Fokker-Planck equation integral dgamma(')tau(gamma('))[ partial differential (gamma('))rho(x,t)]/ partial differential t(gamma('))= integral dmu(')dnu'D(mu('),nu('))[ partial differential (mu('))[rho(x,t)](nu('))]/ partial differential x(mu(')), where tau(gamma(')) and D(mu('),nu(')) are kernels to be chosen; the choice tau(gamma('))=delta(gamma(')-1) and D(mu('),nu('))=delta(mu(')-2)delta(nu(')-1) recovers the normal diffusion equation. We discuss in detail the following cases: (i) a mixture of the porous medium equation, which is connected with nonextensive statistical mechanics, with the normal diffusion equation; (ii) a mixture of the fractional time derivative and normal diffusion equations; (iii) a mixture of the fractional space derivative, which is related with Lévy flights, and normal diffusion equations. In all three cases a crossover is obtained between anomalous and normal diffusions. In cases (i) and (iii), the less diffusive regime occurs for short times, while at long times the more diffusive regime emerges. The opposite occurs in case (ii). The present results could be easily extended to more complex situations (e.g., crossover between two, or even more, different anomalous regimes), and are expected to be useful in the analysis of phenomena where nonlinear and fractional diffusion equations play an important role. Such appears to be the case for isolated long-ranged interaction Hamiltonians, which along time can exhibit a crossover from a longstanding metastable anomalous state to the usual Boltzmann-Gibbs equilibrium one. Another illustration of such crossover occurs in active intracellular transport.

Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations.

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the positivity of the solutions of the proposed equations and establish their relation to the continuous-time random walk theory. We show that the distributed-order time fractional diffusion equation describes the subdiffusion random process that is subordinated to the Wiener process and whose diffusion exponent decreases in time (retarding subdiffusion). This process may lead to superslow diffusion, with the mean square displacement growing logarithmically in time. We also demonstrate that the distributed-order space fractional diffusion equation describes superdiffusion phenomena with the diffusion exponent increasing in time (accelerating superdiffusion).

Anomalous diffusion: nonlinear fractional Fokker–Planck equation

Abstract We discuss the anomalous diffusion associated with a nonlinear fractional Fokker–Planck equation with a diffusion coefficient D∝|x|−θ ( θ∈ R ). Two classes of exact solutions are found. The first one is a modified porous medium equation and corresponds to integer derivatives and a drift force F∝x|x|α−1 ( α∈ R ). The second one corresponds to fractional space derivative in the absence of external drift. The connection with nonextensive statistical mechanics is also discussed in both cases.

Stochastic solution of space-time fractional diffusion equations.

Classical and anomalous diffusion equations employ integer derivatives, fractional derivatives, and other pseudodifferential operators in space. In this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are Mittag-Leffler type. This leads to explicit solutions for space-time fractional diffusion equations with multiscaling space-fractional derivatives, and additional insight into the meaning of these equations.

Lévy flights in external force fields: Langevin and fractional Fokker-Planck equations and their solutions

We consider Levy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Levy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis's q-statistics is studied.

High-order quasi-compact difference schemes for fractional diffusion equations

The continuous time random walk (CTRW) underlies many fundamental processes in non-equilibrium statistical physics. When the jump length of CTRW obeys a power-law distribution, its corresponding Fokker-Planck equation has a space fractional derivative, which characterizes Levy flights. Sometimes the infinite variance of Levy flight discourages it as a physical approach; exponentially tempering the power-law jump length of CTRW makes it more 'physical' and the tempered space fractional diffusion equation appears. This paper provides the basic strategy of deriving the high-order quasi-compact discretizations for the space fractional derivative and the tempered space fractional derivative. The fourth-order quasi-compact discretization for the space fractional derivative is applied to solve a space fractional diffusion equation, and the unconditional stability and convergence of the scheme are theoretically proved and numerically verified. Furthermore, the tempered space fractional diffusion equation is effectively solved by its counterpart, the fourth-order quasi-compact scheme, and the convergence orders are verified numerically.