Finite difference approximations for the fractional Fokker–Planck equation

Chen, Jinjun; Liu, Fawang; Zhuang, Pinghui; Anh, Vo

doi:10.1016/j.apm.2007.11.005

Cited by 213 publications

(106 citation statements)

References 18 publications

Supporting

Mentioning

106

Contrasting

Order By: Relevance

“…(3) with F (x) ≡ 0 is analyzed in an infinite channel by Wang and Guo [7]. At the same time, the numerical methods for fractional partial differential equations have also progressed rapidly, such as finite difference method [8][9][10][11][12][13][14][15], finite element method [16][17][18][19], spectral method [20,21], etc. However, published articles on the numerical methods of FKKEs are very sparse.…”

Section: Introductionmentioning

confidence: 99%

Finite difference methods and their physical constraints for the fractional klein-kramers equation

Deng

Can

2010

Numer. Methods Partial Differential Eq.

View full text Add to dashboard Cite

Incorporating subdiffusive mechanisms into the Klein-Kramers formalism leads to the fractional KleinKramers equation. Then, the equation can effectively describe subdiffusion in the presence of an external force field in the phase space. This article presents the finite difference methods for numerically solving the fractional Klein-Kramers equation and does the detailed stability and error analyses. The stability condition, mv 2 R β ≤ 16, shows the ratio between the kinetic energy of the particle and the temperature of the fluid can not be too large, which well agrees with the physical property of the subdiffusive particle, we call it "physical constraint." The numerical examples are provided to verify the theoretical results on rate of convergence. Moreover, we simulate the fractional Klein-Kramers dynamics and the simulation results further confirm the effectiveness of our numerical schemes.

show abstract

Section: Introductionmentioning

confidence: 99%

Finite difference methods and their physical constraints for the fractional klein-kramers equation

Deng

Can

2010

Numer. Methods Partial Differential Eq.

View full text Add to dashboard Cite

show abstract

“…Cao et al [3] adopted a similar approach for (1.2) and solved the resulting system of fractional ODEs using a second order, backward Euler scheme. Chen et al [4] studied the stability and convergence properties of three implicit finite difference techniques, in each of which the diffusion term was approximated by the standard second order difference approximation at the advanced time level. In related work, Jiang [11] established monotonicity properties of the numerical solutions obtained by using these schemes, and so showed that the time-stepping preserves non-negativity of the solution.…”

Section: γ(α)mentioning

confidence: 99%

Numerical Solution of the Time-Fractional Fokker--Planck Equation with General Forcing

Le¹,

McLean²,

Mustapha³

2016

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. We study two schemes for a time-fractional Fokker-Planck equation with spaceand time-dependent forcing in one space dimension. The first scheme is continuous in time and is discretized in space using a piecewise-linear Galerkin finite element method. The second is continuous in space and employs a time-stepping procedure similar to the classical implicit Euler method. We show that the space discretization is second-order accurate in the spatial L 2 -norm, uniformly in time, whereas the corresponding error for the time-stepping scheme is O(k α ) for a uniform time step k, where α ∈ (1/2, 1) is the fractional diffusion parameter. In numerical experiments using a combined, fully-discrete method, we observe convergence behaviour consistent with these results.

show abstract

“…Our idea is to apply the Laguerre collocation method to discretize (1) to get a linear system of ordinary differential equations (ODEs) thus greatly simplifying the problem, and use the finite difference method (FDM) ( [15]- [18]) to solve the resulting system.…”

Section: Tt XX U X T D X T U X T S X T = +mentioning

confidence: 99%

Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

Sweilam¹,

Khader²,

Adel³

2015

View full text Add to dashboard Cite

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

show abstract

Finite difference approximations for the fractional Fokker–Planck equation

Cited by 213 publications

References 18 publications

Finite difference methods and their physical constraints for the fractional klein-kramers equation

Finite difference methods and their physical constraints for the fractional klein-kramers equation

Numerical Solution of the Time-Fractional Fokker--Planck Equation with General Forcing

Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

Contact Info

Product

Resources

About