Time-fractional Fokker Planck Equation Research Articles

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.

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.

Well-posedness and simulation of weak solutions to the time-fractional Fokker–Planck equation with general forcing

Well-posedness and simulation of weak solutions to the time-fractional Fokker–Planck equation with general forcing

Numerical simulation of the time-fractional Fokker–Planck equation and applications to polymeric fluids

We introduce a new approach to the numerical approximation of the time-fractional Navier–Stokes–Fokker–Planck (TFNSFP) system, which involves the coupling of the incompressible Navier–Stokes equations to a time-fractional Fokker–Planck equation. The model arises in the context of dilute polymeric fluids, and it enhances the standard integer-derivative version of the model by including memory effects. The key challenge associated with the numerical solution of the TFNSFP system is that, in addition to it being nonlocal in time, it is, even in its simplest form, posed on a spatial domain that is the Cartesian product of two d-dimensional domains, for d∈{2,3}, the d-dimensional flow domain and the d-dimensional configuration space domain. We combine a kernel compression technique based on rational approximation of the integral kernel of the time-fractional derivative with a space-splitting method. By doing so, we transform the time-fractional partial differential equation into a fixed number of integer-order in time partial differential equations. The Fokker–Planck equation posed on the 2d-dimensional domain is decoupled into two d-dimensional problems, and a standard combination of the Hermite spectral method on the configuration space domain and a finite element method on the flow domain is then applied. Finally, we combine our numerical scheme for the time-fractional Fokker–Planck equation (TFFPE) with a standard Navier–Stokes solver. We propose an efficient implementation strategy based on the algebraic structure of the discretized time-fractional Fokker–Planck equation, which significantly reduces the computational cost.

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.

A hybrid algorithm for solving fractional Fokker-Planck equations arising in physics and engineering

In this work, we proposed a hybrid algorithm to approximate the solution of Conformable Fractional Fokker-Planck Equation (CFFPE). This algorithm comprises of unification of two methods named Fractional Wave Transformation Method (FWTM) and Differential Transform Method (DTM). The method is based on two steps. The first step is to reduce the given CFPDEs to corresponding Partial Differential Equations (PDEs). Then, the second step is to solve obtained PDEs iteratively by using DTM. Moreover, the algorithm’s efficiency is shown by employing the method successfully to conformable time-fractional Fokker-Planck equation arising in surface physics, plasma physics, polymer physics, laser physics, biophysics, engineering, neurosciences, nonlinear hydrodynamics, population dynamics, pattern formation and marketing. As a result, the obtained data demonstrate that the algorithm is reliable and applicable.

A second-order accurate numerical scheme for a time-fractional Fokker–Planck equation

Abstract A second-order accurate time-stepping scheme for solving a time-fractional Fokker–Planck equation of order $\alpha \in (0, 1)$, with a general driving force, is investigated. A stability bound for the semidiscrete solution is obtained for $\alpha \in (1/2,1)$ via a novel and concise approach. Our stability estimate is $\alpha $-robust in the sense that it remains valid in the limiting case where $\alpha $ approaches $1$ (when the model reduces to the classical Fokker–Planck equation), a limit that presents practical importance. Concerning the error analysis, we obtain an optimal second-order accurate estimate for $\alpha \in (1/2,1)$. A time-graded mesh is used to compensate for the singular behavior of the continuous solution near the origin. The time-stepping scheme scheme is associated with a standard spatial Galerkin finite element discretization to numerically support our theoretical contributions. We employ the resulting fully discrete computable numerical scheme to perform some numerical tests. These tests suggest that the imposed time-graded meshes assumption could be further relaxed, and we observe second-order accuracy even for the case $\alpha \in (0,1/2]$, that is, outside the range covered by the theory.

The existence of mild and classical solutions for time fractional Fokker–Planck equations

The existence of mild and classical solutions for time fractional Fokker–Planck equations

New semi-analytical solutions of the time-fractional Fokker–Planck equation by the neural network method

This paper suggests a neural network method for solving the time-fractional Fokker–Planck equation. An energy function is constructed by means of initial–boundary value conditions. The Caputo derivative is approximated by L1 numerical scheme and an unconstrained discretization minimization problem is presented. Gradient descent algorithm is adopted for neural network training. Semi-analytical solutions are obtained where two numerical examples demonstrate the method’s efficiency.

Method of variable separation for investigating exact solutions and dynamical properties of the time-fractional Fokker–Planck equation

In this paper, the traditional separation method of variables is improved by fixing a part of the variables called the separation method of semi-fixed variables. By using this improved method, the time-fractional Fokker–Planck equation with external force field is studied. Some new exact solutions and dynamical properties of the equation are investigated in various external potential functions such as linear potential, harmonic potential, logarithmic potential, exponential potential, and quartic potential. Some interesting dynamical behaviors and phenomena are discovered. The profiles of several representative exact solutions are illustrated by 3D-graphs and 2D-graphs

A predictor–corrector scheme for solving the time fractional Fokker–Planck equation with uniform and non-uniform meshes

A predictor–corrector scheme for solving the time fractional Fokker–Planck equation with uniform and non-uniform meshes

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

<p style='text-indent:20px;'>In this paper, we study the numerical approximation of a system of PDEs which arises from an optimal control problem for the time-fractional Fokker-Planck equation with time-dependent drift. The system is composed of a backward time-fractional Hamilton-Jacobi-Bellman equation and a forward time-fractional Fokker-Planck equation. We approximate Caputo derivatives in the system by means of L1 schemes and the Hamiltonian by finite differences. The scheme for the Fokker-Planck equation is constructed in such a way that the duality structure of the PDE system is preserved on the discrete level. We prove the well-posedness of the scheme and the convergence to the solution of the continuous problem.</p>

Exact solutions and numerical simulations of time-fractional Fokker-Plank equation for special stochastic process

‎In this paper, a type of time-fractional Fokker-Planck equation (FPE) of the OrnsteinUhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for finding reduced form and exact solutions of the equation. A numerical simulation based on the M¨untz-Legendre polynomials is applied in order to find some approximated solutions of the equation.

Numerical solutions for solving model time‐fractionalFokker–Planckequation

AbstractIn this work, we use two different techniques to discuss approximate analytical solutions for the time‐fractional Fokker–Planck equation (TFFPE), namely the new iterative method (NIM) and the fractional power series method (FPSM). Stability analyses and truncation errors are studied using a procedure like the fundamental von Neumann stability analysis. Discretization is carried out numerically for TFFPE by the implicit finite difference and the Crank–Nicolson method. The techniques used in solving the TFFPE are simple and powerful enough to understand the numerical solutions of linear and nonlinear fractional differential equations. We discuss the approximate solutions obtained using the NIM and FPSM. This is explained by employing tables and shapes. The approximate solutions strongly converge to an accurate solution. All computations in this work were carried out using Maple 16.

A novel approach for the analytical solution of nonlinear time-fractional differential equations

PurposeThe purpose of this paper is to suggest the solution of time-fractional Fornberg–Whitham and time-fractional Fokker–Planck equations by using a novel approach.Design/methodology/approachFirst, some basic properties of fractional derivatives are defined to construct a novel approach. Second, modified Laplace homotopy perturbation method (HPM) is constructed which yields to a direct approach. Third, two numerical examples are presented to show the accuracy of this derived method and graphically results showed that this method is very effective. Finally, convergence of HPM is proved strictly with detail.FindingsIt is not necessary to consider any type of assumptions and hypothesis for the development of this approach. Thus, the suggested method becomes very simple and a better approach for the solution of time-fractional differential equations.Originality/valueAlthough many analytical methods for the solution of fractional partial differential equations are presented in the literature. This novel approach demonstrates that the proposed approach can be applied directly without any kind of assumptions.

An efficient high-order numerical algorithm for the time fractional Fokker–Planck equations

In this paper, a high-order numerical algorithm is derived for a class of time fractional Fokker–Planck equations. To avoid discrete approximation of convection term, we first transform the original equation into an equivalent form, then the spatial derivative and time Riemann–Liouville derivative are approximated by a fourth-order compact formula and a second-order midpoint formula, respectively. The stability and convergence of the developed difference scheme is studied by the energy method. Finally, numerical experiments are provided to demonstrate the effectiveness of the algorithm.

Toward computational algorithm for time-fractional Fokker–Planck models

This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.

The finite volume scheme preserving maximum principle for two-dimensional time-fractional Fokker–Planck equations on distorted meshes

In this paper, we develop a nonlinear finite volume scheme preserving maximum principle for 2D time fractional Fokker–Planck equations on distorted meshes. The characteristic of our method is that it satisfies the discrete maximum principle such that it keeps physical boundedness such as concentration, temperature and density, etc. The analysis is based on an adaptive approach of choosing stencil to construct a maximum-principle-preserving discrete normal flux for diffusive flux. For the advection term, we use the second-order upwind method with proper slope limiter. The fractional derivative is approximated through L1-scheme. The advantages of our scheme are that it is locally conservative and can be applied to distorted meshes with no severe constraint on the time step. Numerical results verify the theoretical result and show that our scheme can preserve discrete maximum principle.

From the space–time fractional integral of the continuous time random walk to the space–time fractional diffusion equations, a short proof and simulation

In this paper, I prove the relation between the integral equation of the continuous time random walk (CTRW) and the space–time fractional diffusion equations by using the definitions of the Weyl fractional derivatives and integrals. I generalize a transformation theorem between the independent variables of the solution of the space–time fractional diffusion equation (stfde) and the solution of the space–time fractional Fokker–Planck equation(stffpe). I simulate the symmetric and the non symmetric random walks for the two mentioned models. I use the asymptotic behavior of the Mittag-Leffler function for generating the waiting time random variable and use the Lévy flight distribution function to generate the jump width random variable for the both models. The simulation results are investigated for different values of the space and time fractional orders.