Numerical Method for The Time Fractional Fokker-Planck Equation

2012 ◽  
Vol 4 (06) ◽  
pp. 848-863 ◽  
Author(s):  
Xue-Nian Cao ◽  
Jiang-Li Fu ◽  
Hu Huang
Keyword(s):  
Theoretical Analysis ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Truncation Error ◽  
Planck Equation ◽  
Local Truncation Error ◽  
Order Of Accuracy ◽  
Fokker Planck Equation ◽  
The Stability ◽  
Fokker Planck

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.

A method for the calculation of characteristics for the solution to stochastic differential equations

2017 ◽  
Vol 23 (3) ◽  
Author(s):  
Alexander Egorov ◽  
Victor Malyutin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Probability Function ◽  
Transition Probability ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Transition Probability Function ◽  
Fokker Planck

AbstractIn this work, a new numerical method to calculate the characteristics of the solution to stochastic differential equations is presented. This method is based on the Fokker–Planck equation for the transition probability function and the representation of the transition probability function by means of eigenfunctions of the Fokker–Planck operator. The results of the numerical experiments are presented.

scholarly journals A Phase-Fitted and Amplification-Fitted Explicit Runge–Kutta–Nyström Pair for Oscillating Systems

2021 ◽  
Vol 26 (3) ◽  
pp. 59
Author(s):  
Musa Ahmed Demba ◽  
Higinio Ramos ◽  
Poom Kumam ◽  
Wiboonsak Watthayu
Keyword(s):  
Stability Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Truncation Error ◽  
Local Truncation Error ◽  
Nyström Methods ◽  
Oscillating Systems ◽  
The Common ◽  
Runge Kutta ◽  
The Stability

An optimized embedded 5(3) pair of explicit Runge–Kutta–Nyström methods with four stages using phase-fitted and amplification-fitted techniques is developed in this paper. The new adapted pair can exactly integrate (except round-off errors) the common test: y″=−w2y. The local truncation error of the new method is derived, and we show that the order of convergence is maintained. The stability analysis is addressed, and we demonstrate that the developed method is absolutely stable, and thus appropriate for solving stiff problems. The numerical experiments show a better performance of the new embedded pair in comparison with other existing RKN pairs of similar characteristics.

A new analysis of a numerical method for the time-fractional Fokker–Planck equation with general forcing

2019 ◽  
Vol 40 (2) ◽  
pp. 1217-1240 ◽  
Author(s):  
Can Huang ◽  
Kim Ngan Le ◽  
Martin Stynes
Keyword(s):  
Numerical Method ◽  
Blow Up ◽  
Planck Equation ◽  
Stability And Convergence ◽  
Time Stability ◽  
Backward Euler Scheme ◽  
Fokker Planck Equation ◽  
Backward Euler ◽  
Numer Anal ◽  
Fokker Planck

Abstract First, a new convergence analysis is given for the semidiscrete (finite elements in space) numerical method that is used in Le et al. (2016, Numerical solution of the time-fractional Fokker–Planck equation with general forcing. SIAM J. Numer. Anal.,54 1763–1784) to solve the time-fractional Fokker–Planck equation on a domain $\varOmega \times [0,T]$ with general forcing, i.e., where the forcing term is a function of both space and time. Stability and convergence are proved in a fractional norm that is stronger than the $L^2(\varOmega )$ norm used in the above paper. Furthermore, unlike the bounds proved in Le et al., the constant multipliers in our analysis do not blow up as the order of the fractional derivative $\alpha $ approaches the classical value of $1$. Secondly, for the semidiscrete (L1 scheme in time) method for the same Fokker–Planck problem, we present a new $L^2(\varOmega )$ convergence proof that avoids a flaw in the analysis of Le et al.’s paper for the semidiscrete (backward Euler scheme in time) method.

scholarly journals Interval Shannon Wavelet Collocation Method for Fractional Fokker-Planck Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Shu-Li Mei ◽  
De-Hai Zhu
Keyword(s):  
Numerical Method ◽  
Boundary Effect ◽  
Heat Bath ◽  
Planck Equation ◽  
Difference Approximation ◽  
Central Difference ◽  
Fokker Planck Equation ◽  
Nonlinear Force ◽  
Shannon Wavelet ◽  
Fokker Planck

Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In this paper, we present an interval Shannon wavelet numerical method for the FFPE. In this method, a new concept named “dynamic interval wavelet” is proposed to solve the problem that the numerical solution of the fractional PDE is usually sensitive to boundary conditions. Comparing with the traditional wavelet defined in the interval, the Newton interpolator is employed instead of the Lagrange interpolation operator, so, the extrapolation points in the interval wavelet can be chosen dynamically to restrict the boundary effect without increase of the calculation amount. In order to avoid unlimited increasing of the extrapolation points, both the error tolerance and the condition number are taken as indicators for the dynamic choice of the extrapolation points. Then, combining with the finite difference technology, a new numerical method for the time fractional partial differential equation is constructed. A simple Fokker-Planck equation is taken as an example to illustrate the effectiveness by comparing with the Grunwald-Letnikov central difference approximation (GL-CDA).

A Numerical Method for Two-Dimensional Fokker-Planck Equation

1982 ◽  
Vol 1 (4) ◽  
pp. 507-513
Author(s):  
Yang Ying-nul ◽  
Wu Xi-zhen ◽  
Peng Qing-quan
Keyword(s):  
Numerical Method ◽  
Planck Equation ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Fokker Planck

scholarly journals Numerical Solution of Two Dimensional Time-Space Fractional Fokker Planck Equation With Variable Coefficients

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1260
Author(s):  
Elsayed I. Mahmoud ◽  
Viktor N. Orlov
Keyword(s):  
Brownian Particle ◽  
Planck Equation ◽  
Variable Coefficients ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
Fokker Planck Equation ◽  
Time Space ◽  
Liouville Derivative ◽  
The Stability ◽  
Fokker Planck

This paper presents a practical numerical method, an implicit finite-difference scheme for solving a two-dimensional time-space fractional Fokker–Planck equation with space–time depending on variable coefficients and source term, which represents a model of a Brownian particle in a periodic potential. The Caputo derivative and the Riemann–Liouville derivative are considered in the temporal and spatial directions, respectively. The Riemann–Liouville derivative is approximated by the standard Grünwald approximation and the shifted Grünwald approximation. The stability and convergence of the numerical scheme are discussed. Finally, we provide a numerical example to test the theoretical analysis.

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