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.

Brownian Motion of Rotating Particles

1968 ◽  
Vol 23 (4) ◽  
pp. 597-609 ◽  
Author(s):  
Siegfried Hess
Keyword(s):  
Brownian Motion ◽  
Angular Velocity ◽  
Brownian Particle ◽  
Planck Equation ◽  
Transverse Force ◽  
Particle Collision ◽  
Fokker Planck Equation ◽  
Rotational Motions ◽  
Transport Relaxation ◽  
Fokker Planck

A kinetic theory for the Brownian motion of spherical rotating particles is given starting from a generalized Fokker-Planck equation. The generalized Fokker-Planck collision operator is a sum of two ordinary Fokker-Planck differential operators in velocity and angular velocity space respectively plus a third term which provides a coupling of translational and rotational motions. This term stems from a transverse force proportional to the cross product of velocity and angular velocity of a Brownian particle. Collision brackets pertaining to the generalized Fokker-Planck operator are defined and their general properties are discussed. Application of WALDMANN'S moment method to the Fokker-Planck equation yields a set of coupled linear differential equations (transport-relaxation equations) for certain local mean values. The constitutive laws for diffusion, heat conduction by Brownian particles and spin diffusion are deduced from the transport-relaxation equations. The transport-relaxations coefficients appearing in them are given in terms of the two friction coefficients for the damping of translational and rotational motions and a third coefficient which is a measure of the transverse force. By the coupling of translational and rotational motions a diffusion flow gives rise to a correlation of linear and angular velocities.

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.

Dynamic Snow Distribution Modeling using the Fokker-Planck Equation Approach

2021 ◽  
Author(s):  
Noriaki Ohara
Keyword(s):  
Time Series ◽  
Snow Depth ◽  
Planck Equation ◽  
Snow Accumulation ◽  
Airborne Lidar ◽  
Equation Approach ◽  
Diffusion Type ◽  
Fokker Planck Equation ◽  
Time Space ◽  
Fokker Planck

<p>The Fokker-Planck equation (FPE) describes the time evolution of the distribution function of fluctuating macroscopic variables.  Although the FPE was originally derived for the Brownian motion, this framework can be applied to various physical processes.  In this presentation, applications in the snow accumulation and thaw process, which attributes to considerable spatial and temporal variations, are discussed. It is well known that snow process is a major source of heterogeneity in hydrological systems in high altitude or latitude regions; therefore, better treatment of the snow sub-grid variability is desirable. The main advantage of the FPE approach is that it can dynamically compute the probability density function (PDF) governed by an advection-diffusion type FPE without a prescribed PDF.</p><p>First, a bivariate FPE was derived from point scale process-based governing equations (Ohara et al., 2008). This FPE can express the evolution of the PDF of snow depth and temperature within a finite space, possibly a computational cell or small basin, whose shape is irrelevant. This conceptual model was proven to be effective through comparing to the corresponding Monte-Carlo simulation.  Then, the more realistic single variated FPE model for snow depth was implemented with the snow redistribution and snowmelt rate as the main sources of stochasticity. In this study, several realistic approximations were proposed to compute the time-space covariances describing effects induced by uneven snowmelt and snow redistribution.</p><p>Meanwhile, observed high-resolution snow depth data was analyzed using statistical methods to characterize the sub-grid variability of snow depth, which is essential to validate the FPE model for representing such sub-grid variability.  Airborne light detection and ranging (Lidar) provided the snow depth measurements at 0.5 m resolution over two mountainous areas in southwestern Wyoming, Snowy Range and Laramie Range (He et al., 2019). It was found that PDFs of snow depth tend to be Gaussian distributions in the forest areas. However, due to the no-snow areas effect, mainly caused by snow redistribution and uneven snowmelt, the PDFs are eventually skewed as non-Gaussian distribution.</p><p>The simulated results of the FPE model were validated using the measured time series of snow depth at one site and the spatial distributions of snow depth measured by ground penetrating radar (GPR) and airborne Lidar. The modeled and observed time series of the mean snow depth agreed very well while the simulated PDFs of snow depth within the study area were comparable to the observed PDFs of snow depth by GPR and Lidar (He and Ohara, 2019). Accordingly, the FPE model is capable to capture the main characteristics of the snow sub-grid variability in the nature.</p><p><strong>References</strong></p><p>Ohara, N., Kavvas, M. L., & Chen, Z. Q. (2008). Stochastic upscaling for snow accumulation and melt processes with PDF approach. Journal of Hydrologic Engineering, 13(12), 1103-1118.</p><p>He, S., Ohara, N., & Miller, S. N. (2019). Understanding subgrid variability of snow depth at 1‐km scale using Lidar measurements. Hydrological Processes, 33(11), 1525-1537.</p><p>He, S., & Ohara, N. (2019). Modeling subgrid variability of snow depth using the Fokker‐Planck equation approach. Water Resources Research, 55(4), 3137-3155.</p>

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