Fractional Fokker-Planck Equation Research Articles

Collocation Finite Element Method for the Fractional Fokker–Planck Equation

ABSTRACTIn this study, the approximate results of the fractional Fokker–Planck equations have been investigated. First, finite element schemes have been obtained using collocation finite element method based on the trigonometric quintic B‐spline basis functions. Then, the present method is tested on two fundamental problems having appropriate initial conditions. The newly obtained numerical results contained the error norms and for various temporal and spatial steps are compared with the exact ones and other solutions. More accurate results have been obtained for large numbers of spatial and temporal elements.

Enhanced fifth-kind Chebyshev polynomial Petrov–Galerkin algorithm for time-fractional Fokker–Planck equation

This paper employs a numerical method for the numerical treatment of the time fractional Fokker–Planck equation. This method is based on applying the spectral Petrov–Galerkin method. We utilize suitable combinations for the shifted Chebyshev polynomial of the fifth-kind as basis functions. The key idea of the suggested strategy is to transform the governed boundary-value problem into a set of linear algebraic equations using the spectral Petrov–Galerkin method. Several approaches are available to solve the resultant linear system. An in-depth investigation is conducted on the convergence and error analysis of the expansion of the shifted Chebyshev polynomial function of the fifth kind. Many examples are provided to illustrate the precision of the suggested approach.

Semi analytical scheme for the presentation of solution to Fractional Fokker–Planck Equation

In this work, we have also proposed new semi analytical scheme combining least-squares and asymptotic homotopy perturbation methods AHPM for the solution of Fractional Fokker–Planck Equation (FFPE). The Caputo version of derivative is used. The finding of this work is fast convergence of the AHPM to the solutions of linear and nonlinear applied problems. AHPM is a modified form of the HPM and is particularly useful for solving problems with small or large parameters. This can be seen as a clear advantage of this new semi analytical method over existing methods because its implementation and the calculation involved in it are much simpler and easier. The proposed method solved the problems without using Ji-Huan He and Adomian polynomials. Numerical solutions obtained in this way show that this approach is simple and computationally very attractive. An error estimate for the solution is also provided. Approximate solutions of AHPM are tabulated and graphically displayed to show that AHPM is effective and accurate.

Fractal-view analysis of local fractional Fokker–Planck equation occurring in modelling of particle’s Brownian motion

Fractal-view analysis of local fractional Fokker–Planck equation occurring in modelling of particle’s Brownian motion

An operational matrix strategy for time fractional Fokker-Planck equation in an unbounded space domain

In this study, the Caputo fractional derivative is used to define time fractional Fokker-Planck equation in an unbounded domain. To solve this equation, the Jacobi polynomials together with the tanh-Jacobi functions are employed. The operational matrices of the classical and fractional derivatives of these basis functions are obtained to use them in constructing a numerical method for the expressed equation. In the proposed method, the introduced basis functions are used simultaneously to approximate the equation’s unknown solution. More precisely, the shifted Jacobi polynomials are applied to approximate the solution in the temporal direction and the tanh-Jacobi functions are utilized to approximate the solution in the spatial direction. By substitute the expressed approximation into the equation and employing the introduced operational matrix, solving the problem under consideration transforms into solving an algebraic system of equations, which can be solved easily. The accuracy and efficiency of the presented method are investigated numerically by solving some numerical examples. The reported results confirms the high accuracy of the established method.

Application of Laplace Adomian decomposition method for fractional Fokker-Planck equation and time fractional coupled Boussinesq-Burger equations

PurposeFractional Fokker-Planck equation (FFPE) and time fractional coupled Boussinesq-Burger equations (TFCBBEs) play important roles in the fields of solute transport, fluid dynamics, respectively. Although there are many methods for solving the approximate solution, simple and effective methods are more preferred. This paper aims to utilize Laplace Adomian decomposition method (LADM) to construct approximate solutions for these two types of equations and gives some examples of numerical calculations, which can prove the validity of LADM by comparing the error between the calculated results and the exact solution.Design/methodology/approachThis paper analyzes and investigates the time-space fractional partial differential equations based on the LADM method in the sense of Caputo fractional derivative, which is a combination of the Laplace transform and the Adomian decomposition method. LADM method was first proposed by Khuri in 2001. Many partial differential equations which can describe the physical phenomena are solved by applying LADM and it has been used extensively to solve approximate solutions of partial differential and fractional partial differential equations.FindingsThis paper obtained an approximate solution to the FFPE and TFCBBEs by using the LADM. A number of numerical examples and graphs are used to compare the errors between the results and the exact solutions. The results show that LADM is a simple and effective mathematical technique to construct the approximate solutions of nonlinear time-space fractional equations in this work.Originality/valueThis paper verifies the effectiveness of this method by using the LADM to solve the FFPE and TFCBBEs. In addition, these two equations are very meaningful, and this paper will be helpful in the study of atmospheric diffusion, shallow water waves and other areas. And this paper also generalizes the drift and diffusion terms of the FFPE equation to the general form, which provides a great convenience for our future studies.

A space-time Galerkin Müntz spectral method for the time fractional Fokker–Planck equation

In this paper, we propose a space-time Galerkin spectral method for the time fractional Fokker–Planck equation. This approach is based on combining temporal Müntz Jacobi polynomials spectral method with spatial Legendre polynomials spectral method. Based on the well-posedness and regularity for the re-scaled problem of a linear model problem which reflects the main difficulty for solving the equivalent equation (i.e. the time fractional convection-diffusion equation): the singularity of the solution in time, we explain in detail why we use the Müntz polynomials to approximate in time. The well-posedness and stability of the discrete scheme as well as its continuous problem are established. Moreover, the error estimation of the space-time approach is derived. We find that the proposed method can attain spectral accuracy regardless of whether the solution of the original equation is smooth or non-smooth. Numerical experiments substantiate the theoretical results.

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Numerical methods for fractional Fokker–Planck equation with multiplicative Marcus Lévy noises

The Fokker–Planck equation (FPE) is an important deterministic tool for investigating stochastic dynamical systems. In this paper, we consider the space-time fractional FPE driven by multiplicative Marcus Lévy noises. Efficient numerical schemes are presented to solve the equations. Stability and convergence of the methods are also discussed. We give some numerical experiments to validate our schemes, and examine the effects of parameters on solutions. Additionally, we analyze the maximal likely trajectories and the critical time for the change of the most probability location.

The Generalized Hermite Spectral Method Combined with Laplace Transform for Solution of the Time Fractional Fokker–Planck Equation

The paper proposes a method to solve the time fractional Fokker–Planck equation using the generalized Hermite spectral method and Laplace transform. By the Laplace transform, the ordinary differential equation about the orthogonal expansion coefficient is transformed into a linear algebraic equation system, with the coefficient matrix being a tridiagonal lower triangular matrix. The orthogonal expansion coefficients of frequency domain components are obtained through this transformation. The approximate solution at any time is then obtained by using the numerical inverse Laplace transform with the Talbot algorithm. Numerical experiments have been carried out to demonstrate the high accuracy and efficiency of the proposed method.

Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations

Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations

Circular cumulant reductions for macroscopic dynamics of oscillator populations with non-Gaussian noise.

We employ the circular cumulant approach to construct a low dimensional description of the macroscopic dynamics of populations of phase oscillators (elements) subject to non-Gaussian white noise. Two-cumulant reduction equations for α-stable noises are derived. The implementation of the approach is demonstrated for the case of the Kuramoto ensemble with non-Gaussian noise. The results of direct numerical simulation of the ensemble of N=1500 oscillators and the "exact" numerical solution for the fractional Fokker-Planck equation in the Fourier space are found to be in good agreement with the analytical solutions for two feasible circular cumulant model reductions. We also illustrate that the two-cumulant model reduction is useful for studying the bifurcations of chimera states in hierarchical populations of coupled noisy phase oscillators.

Jacobian spectral collocation method for spatio-temporal coupled Fokker-Planck equation with variable-order fractional derivative

Fractional Fokker–Planck equation plays an important role in describing anomalous dynamics. In this paper, we consider a spectral method for the spatio-temporal coupled fractional Fokker-Planck equation with variable-order fractional derivative. After introducing a suitable intermediate variable, the original equation can be split into two parts and decoupled. Then a spectral collocation method is employed to the resulting equation. Specifically, the points collocated both in temporal and spatial domains are related to fractional Jacobi functions, and the so-called fractional interpolation functions are used as basis functions for temporal semi-discretization. A recurrence relation is derived for fast generation of the fractional differentiation matrix. Regularity analysis for the solution as well as the convergence estimates for the spectral approximation are established for the constant-order case. Several numerical tests demonstrate the efficiency of the method and support the theoretical results.

Derivation of the Fractional Fokker–Planck Equation for Stable Lévy with Financial Applications

This paper aims to propose a generalized fractional Fokker–Planck equation based on a stable Lévy stochastic process. To develop the general fractional equation, we will use the Lévy process rather than the Brownian motion. Due to the Lévy process, this fractional equation can provide a better description of heavy tails and skewness. The analytical solution is chosen to solve the fractional equation and is expressed using the H-function to demonstrate the indicator entropy production rate. We model market data using a stable distribution to demonstrate the relationships between the tails and the new fractional Fokker–Planck model, as well as develop an R code that can be used to draw figures from real data.

1/f noise and anomalous scaling in Lévy noise-driven on–off intermittency

On–off intermittency occurs in nonequilibrium physical systems close to bifurcation points, and is characterised by an aperiodic switching between a large-amplitude ‘on’ state and a small-amplitude ‘off’ state. Lévy on–off intermittency is a recently introduced generalisation of on–off intermittency to multiplicative Lévy noise, which depends on a stability parameter α and a skewness parameter β. Here, we derive two novel results on Lévy on–off intermittency by leveraging known exact results on the first-passage time statistics of Lévy flights. First, we compute anomalous critical exponents explicitly as a function of arbitrary Lévy noise parameters for the first time, by a heuristic method, extending previous results. The predictions are verified using numerical solutions of the fractional Fokker–Planck equation. Second, we derive the power spectrum S(f) of Lévy on–off intermittency, and show that it displays a power law at low frequencies f, where depends on the noise parameters . An explicit expression for κ is obtained in terms of . The predictions are verified using long time series realisations of Lévy on–off intermittency. Our findings help shed light on instabilities subject to non-equilibrium, power-law-distributed fluctuations, emphasizing that their properties can differ starkly from the case of Gaussian fluctuations.

A SPACE-TIME SPECTRAL COLLOCATION METHOD FOR SOLVING THE VARIABLE-ORDER FRACTIONAL FOKKER-PLANCK EQUATION

A numerical approach for solving the variable-order fractional FokkerPlanck equation (VO-FFPE) is proposed. The computational scheme is based on the shifted Legendre Gauss-Lobatto and the shifted Chebyshev GaussRadau collocation methods. The VO-FFPE is written as a truncated series of shifted Legendre and shifted Chebyshev polynomials for space and time variables, respectively. The residuals of the VO-FFPE at the shifted Legendre Gauss-Lobatto and shifted Chebyshev Gauss-Radau quadrature points are estimated. The original problem is converted into a system of algebraic equations that can be solved easily. Several examples are presented to demonstrate the efficacy of the technique.

A transformed $ L1 $ Legendre-Galerkin spectral method for time fractional Fokker-Planck equations

<abstract><p>The numerical solutions of time $ \alpha $-order $ (\alpha \in (0, 1)) $ Caputo fractional Fokker-Planck equations is considered. The constructed method is consist of the transformed $ L1 $ ($ TL1 $) scheme in the temporal direction and the Legendre-Galerkin spectral method in the spatial direction. It has been shown that the $ TL1 $ Legendre-Galerkin spectral method in $ L^2 $-norm is exponential order convergent in space and ($ 2-\alpha $)-th order convergent in time. Several numerical examples are given to verify the obtained theoretical results.</p></abstract>

Dimension dependent properties of subdiffusions in damping force fields from an inference perspective

We investigate the fractional Fokker-Planck equation subject to a damping force with an emphasis on its dimension dependent properties. We reveal a variety of surprising properties of its solution through the lens of the probability density function of the corresponding stochastic process with nonlinear mean square displacements, such as existence, singularity, regularity, modality, stationarity and second-order structure, which are largely dependent on the dimension and the random clock. Taking into account that the trajectory information is most often collected from multidimensional systems, the discovered facts have the potential to play important roles as key foundations and alerts for inference, model identification and prediction, when departing from the well-understood univariate framework.

Shehu Variational Iteration Method For Solve Some Fractional Differential Equations

The idea proposed in the work is to extend the Shehu transform method to resolve the nonlinear fractional partial differential equations by combining them with the variational iteration method (VIM). We apply this technique to solve nonlinear fractional equations as nonlinear time fractional Fokker Planck equation.

Meshless local Petrov-Galerkin method for 2D fractional Fokker-Planck equation involved with the ABC fractional derivative

This paper examines a time fractional version of the 2D Fokker-Planck equation involved with the Atangana-Baleanu-Caputo fractional derivative, under the Dirichlet boundary conditions. A fully discretization approach based on the meshless local Petrov-Galerkin method and ( 3 − β ) -order approximation is proposed for this equation. More precisely, we apply the meshless local Petrov-Galerkin method based on the Moving Kriging interpolation to discretize the space domain, and utilize the ( 3 − β ) -order approximation together with the θ -weighted finite difference method to discretize the temporal domain. By implementing this method, we get a solution for the problem by solving a system of algebraic equations. The validity of the method is investigated by solving four examples.