Fractional Fokker-Planck Equation

Baumann, Gerd; Stenger, Frank

doi:10.3390/math5010012

Cited by 15 publications

(5 citation statements)

References 30 publications

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“…whereL(x) is the time-independent FP operator (for example, − ∂ ∂x F (x) + D ∂ 2 ∂x 2 with a force F ). In a general case, the operatorL can be both multidimensional and fractional in space [11,12]. The operational time τ = S α (t ) is obtained from a strictly increasing α-stable Lévy process T α (τ ) which is nothing else but the continuous limit of a sequence T i , i = 1, 2, .…”

Section: Simple Power-law Kernelmentioning

confidence: 99%

Control of the transient subdiffusion exponent at short and long times

Stanislavsky

Weron

2019

Phys. Rev. Research

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We consider a simple, physically and biologically motivated model of transient subdiffusion having different exponents in the mean squared displacement at short and long times. For this purpose, the sum of two independent stable processes (with different indexes) as subordinators is used. Based on their infinitely divisible properties, the probability density function of particle displacements and the Fokker-Planck-type equation are obtained. Manifesting different origins of inhomogeneities (traps) in the system, the subordinators' indexes affect the asymptotic behavior of the mean squared displacement for a particle walking in the environment. Links to the Jonscher universal relaxation law and the Prabhakar-type memory functions are given.

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Section: Simple Power-law Kernelmentioning

confidence: 99%

Control of the transient subdiffusion exponent at short and long times

Stanislavsky

Weron

2019

Phys. Rev. Research

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“…Results for the discrete L 2 − norm of errors are reported in Table 4. We take the same time step size τ = 0.01( 1 3 L ) 2 , where L is the fine level in the two-grid Algorithm. Second-order accuracy O(h 2 + τ ) is observed in the numerical results for the proposed two-level scheme, see Table 4.…”

Section: Time-dependant Fp Equation With Linear Driftmentioning

confidence: 99%

“…In the year 2020, the research to find the numerical solution to the stochastic models and henece the FP equation is still on; e.g., in [8], a discretization scheme is developed to solve the one-dimensional nonlinear Fokker-Planck-Kolmogorov equation that preserves the nonnegativity of the solution and conserves the mass; a solution to the FokkerPlanck Equation with piecewise-constant drift is proposed in [11], a numerical method, named as information length, for measuring distances between statistical states as represented by PDF has been proposed in [1]. Also, there has been work on Fractional Fokker-Planck Equation as well, e.g., a space-time Petrov-Galerkin spectral method for time fractional FP equation with nonsmooth solution has been studied in [25] and a numerical solution of the Cauchy problem for the fractional FP equation in connection with Sinc convolution methods is proposed in [2].…”

Section: Introductionmentioning

confidence: 99%

Numerical solution for Fokker-Planck equation using a two-level scheme

Butt

2020

Preprint

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A numerical solution to the Fokker-Planck equation using a two-level scheme is presented. The Fokker-Planck (FP) equation is of parabolic type equation govern the time evolution of probability density function of the stochastic processes. The FP equation also preserves the positivity and conservative of the total probability. A Chang-Cooper discretization scheme is used to ensure the positiveness and conservation of the total probability with second-order accuracy. We investigate a two-level scheme with factor-three-coarsening strategy and have a significant reduction in computations and CPU time. Numerical experiments are performed to validate the efficiency and second-order accuracy of the proposed two-level algorithm with backward time-difference schemes.

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“…The Fokker-Planck equation was first studied by Fokker and Planck to model the Brownian motion of particles [9]. This equation, as a well-known parabolic type equation, is widely applied in different fields of sciences, including solid state physics, physics-chemistry, theoretical biology, quantum optics, and orbit theory [10]. Also, this equation is used to model the change of probability in time and space of stochastic functions and to model the transfer of solutes [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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

Heydari,

Azin,

Hosseini

2024

Phys. Scr.

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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.

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Fractional Fokker-Planck Equation

Cited by 15 publications

References 30 publications

Control of the transient subdiffusion exponent at short and long times

Control of the transient subdiffusion exponent at short and long times

Numerical solution for Fokker-Planck equation using a two-level scheme

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

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