Time-fractional Fokker Planck Equation Research Articles

Linear response characteristics of time-dependent time fractional Fokker–Planck equation systems

The time fractional Fokker–Planck equation approach is an important tool for modeling subdiffusion. When the external field is time modulated, two types of time-dependent time fractional Fokker–Planck equations have been proposed, both reduced to the same time-dependent time fractional Fokker–Planck equation when the external field is time uncorrelated. The first type is strictly deduced as the continuous limit of the continuous time random walk with time modulated Boltzmann jumping weight, while the second type is derived by ideally assuming that the jump probabilities can be evaluated at the start of the waiting time prior to jumping. For the first time we obtain the linear response characteristic for the first type of the time fractional Fokker–Planck equation systems, and for a comparison we revisit the corresponding result for the second type of the time fractional Fokker–Planck equation systems, and the similarity and difference between them is discussed with an application example. The investigation not only helps in understanding the competition between subdiffusion and time-dependent modulation, but also has significance in accessing the spectral properties of spontaneous fluctuation and the linear dynamic susceptibility of external perturbation in subdiffusive processes.

Group analysis and exact solutions of the time fractional Fokker–Planck equation

In this paper, the Lie symmetry analysis method is extended to deal with the nonlinear time fractional Fokker–Planck (FP) equation with Riemann–Liouville derivative. The Erdélyi–Kober fractional derivative which is depending on a parameter α, is used for the reduction of FP equation. Symmetry reduction is provided and some exact analytic solutions to the time fractional FP equation are investigated by virtue of the reduction method introduced by M.C. Nucci.

A new analysis of stability and convergence for finite difference schemes solving the time fractional Fokker–Planck equation

This paper presents a new analysis of stability and convergence for finite difference methods used to solve the time fractional Fokker–Planck equation. We show the monotone properties of the numerical solutions with respect to initial values and truncation errors, based on which we prove the stability and convergence under discrete L1 norm.

Analytical Solution of the Time Fractional Fokker-Planck Equation

A nonperturbative approximate analytic solution is derived for the time fractional Fokker-Planck (F-P) equation by using Adomian’s Decomposition Method (ADM). The solution is expressed in terms of Mittag- Leffler function. The present method performs extremely well in terms of accuracy, efficiency and simplicity.

Numerical Treatment of the Modified Time Fractional Fokker-Planck Equation

A numerical method for the modified time fractional Fokker-Planck equation is proposed. Stability and convergence of the method are rigorously discussed by means of the Fourier method. We prove that the difference scheme is unconditionally stable, and convergence order isO(τ+h4), whereτandhare the temporal and spatial step sizes, respectively. Finally, numerical results are given to confirm the theoretical analysis.

Numerical Computation of Time-Fractional Fokker–Planck Equation Arising in Solid State Physics and Circuit Theory Numerical Computation of Time-Fractional Fokker–Planck Equation Arising in Solid State Physics and Circuit Theory

The main aim of the present work is to propose a new and simple algorithm to obtain a quick and accurate analytical solution of the time fractional Fokker-Plank equation which arises in various fields in natural science, including solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. This new and simple algorithm is an innovative adjustment in Laplace transform algorithm which makes the calculations much simpler and applicable to several practical problems in science and engineering. The proposed scheme finds the solutions without any discretization or restrictive assumptions and is free from round-off errors and therefore reduces the numerical computations to a great extent. Furthermore, several numerical examples are presented to illustrate the accuracy and the stability of the method.

Accelerating subdiffusions governed by multiple-order time-fractional diffusion equations: Stochastic representation by a subordinated Brownian motion and computer simulations

We study anomalous diffusion governed by the n-tuple order time-fractional Fokker-Planck equation of Riemann-Liouville type. We find a stochastic representation of this anomalous diffusion; that is, the stochastic process with probability density function evolving according to the considered diffusion equation. The stochastic

Interval Wavelet Numerical Method on Fokker-Planck Equations for Nonlinear Random System

The Fokker-Planck-Kolmogorov (FPK) equation governs the probability density function (p.d.f.) of the dynamic response of a particular class of linear or nonlinear system to random excitation. An interval wavelet numerical method (IWNM) for nonlinear random systems is proposed using interval Shannon-Gabor wavelet interpolation operator. An FPK equation for nonlinear oscillators and a time fractional Fokker-Planck equation are taken as examples to illustrate its effectiveness and efficiency. Compared with the common wavelet collocation methods, IWNM can decrease the boundary effect greatly. Compared with the finite difference method for the time fractional Fokker-Planck equation, IWNM can improve the calculation precision evidently.

Numerical Method for The Time Fractional Fokker-Planck Equation

AbstractIn this paper, a new numerical algorithm for solving the time fractional Fokker-Planck equation is proposed. The analysis of local truncation error and the stability of this method are investigated. Theoretical analysis and numerical experiments show that the proposed method has higher order of accuracy for solving the time fractional Fokker-Planck equation.

Finite difference/predictor–corrector approximations for the space and time fractional Fokker–Planck equation

In this work, by using the properties of the Riemann–Liouville derivative and the Caputo derivative, we firstly transform the space and time fractional, in the sense of the Riemann–Liouville derivative, Fokker–Planck equation to a new fractional PDE with a Caputo time derivative. After discretizing the spatial (classical and fractional) derivatives of the new fractional PDE using a finite difference method, we use the predictor–corrector approach to approximate the FODEs obtained. Conditional stability and convergence of the numerical scheme are rigorously established. We prove that the numerical scheme is stable and that the numerical solution converges to the exact solution with order O ( h + k min { 1 + 2 α , 2 } ) if k α / h μ < C . Numerical experiments are performed to demonstrate the effectiveness of the algorithm and confirm the theoretical claims.

Analytical approach to Fokker–Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method

In this study, we present numerical solutions for the space- and time-fractional Fokker–Planck equation using the homotopy perturbation method (HPM). The fractional derivatives are described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. Some examples are given and comparisons are made, the comparisons show that the homotopy perturbation method is very effective and convenient and overcome the difficulty of traditional methods. The numerical results show that the approaches are easy to implement and accurate when applied to space- and time-fractional Fokker–Planck equations. The methods introduce a promising tool for solving many space–time fractional partial differential equations.

N-dimensional fractional Fokker–Planck equation and its solutions for anomalous radial two-phase flow in porous media

In this paper we present a time fractional Fokker–Planck equation (fFPE) for radial two-phase flow of liquid and gas in porous media. The fFPE of order α is solved for both two- and three-dimensional flow patterns using the Laplace transform method. The general solutions of the fFPE for both two- and three- dimensional flows are given as a convolution integral of the input and a kernel in the Laplace domain. Special solutions for a large value and a periodic boundary condition are also given in the time domain when the inverse Laplace transform can be found analytically. The fFPE for two-phase flow in porous media presented in this paper is the first report of its kind.

Finite Element Method for the Space and Time Fractional Fokker–Planck Equation

We develop the finite element method for the numerical resolution of the space and time fractional Fokker–Planck equation, which is an effective tool for describing a process with both traps and flights; the time fractional derivative of the equation is used to characterize the traps, and the flights are depicted by the space fractional derivative. The stability and error estimates are rigorously established, and we prove that the convergent order is $O(k^{2-\alpha}+h^\mu)$, where k is the time step size and h the space step size. Numerical computations are presented which demonstrate the effectiveness of the method and confirm the theoretical claims.

Winding number of fractional Brownian motion

We find the exact winding number distribution of Riemann–Liouville fractional Brownian motion for large times in two dimensions using the propagator of a free particle. The distribution is similar to the Brownian-motion case and is of Cauchy type. In addition we find the winding number distribution of the fractal-time process, i.e. the time-fractional Fokker–Planck equation, in the presence of a finite-size winding center.

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 + 2 α , 2 } ) + O ( h 2 ) , 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.

Numerical solution of Fokker–Planck equation with space- and time-fractional derivatives

Abstract This Letter presents numerical solutions for the space- and time-fractional Fokker–Planck equation using the variational iteration method and the Adomian decomposition method. The fractional derivatives are described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. Some examples are given and comparisons are made, the comparisons show that the variational iteration method is very effective and convenient and overcome the difficulty arising in calculating Adomian polynomials. The numerical results show that the two approaches are easy to implement and accurate when applied to space- and time-fractional Fokker–Planck equations. The methods introduce a promising tool for solving many space–time fractional partial differential equations.

Brazilian quartz—a strategic mineral : Paul F. Kerr in Mining and Metallurgy, Vol. 23, No. 427

In this paper the transition joint probability density function of the solution of the Ornstein–Uhlenbeck process is presented by a deterministic parabolic time-fractional PDE (FPDE), named time-fractional Fokker–Planck equation. The article generally is divided to two parts: theoretical and approximated analysis. In the theoretical sections Lie group method and invariant subspace method are applied for finding exact solutions and conservation laws of the considered equation. In the next part, by Chebyshev wavelets’s method the numerical solutions are driven. Then the usefulness of this approximated method is comparing with the exact solutions by some plotted graphs.

Wasserstein gradient flow formulation of the time-fractional Fokker–Planck equation

In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [26]. We propose a JKO type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.