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S Chen - One of the best experts on this subject based on the ideXlab platform.
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finite difference approximations for the fractional Fokker Planck Equation
Applied Mathematical Modelling, 2009Co-Authors: S Chen, P. ZhuangAbstract:The fractional Fokker–Planck Equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker–Planck Equation on a finite domain. The solvability, stability, consistency, and convergence of these methods are discussed. Their stability is proved by the energy method. Two numerical examples are also presented to evaluate these finite difference methods against the exact analytical solutions.
Anil D Gangal - One of the best experts on this subject based on the ideXlab platform.
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local fractional Fokker Planck Equation
Physical Review Letters, 1998Co-Authors: Kiran M Kolwankar, Anil D GangalAbstract:We propose a new class of differential Equations, which we call local fractional differential Equations. They involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of the Fokker-Planck Equation has been derived starting from the Chapman-Kolmogorov condition. We solve the Equation with a specific choice of the transition probability and show how subdiffusive behavior can arise.
Eli Barkai - One of the best experts on this subject based on the ideXlab platform.
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Fractional Fokker-Planck Equation, solution, and application
Physical Review E, 2001Co-Authors: Eli BarkaiAbstract:Recently, Metzler et al. Phys. Rev. Lett. 82, 3563 ~1999!, introduced a fractional Fokker-Planck Equation FFPE! describing a subdiffusive behavior of a particle under the combined influence of external nonlinear orce field, and a Boltzmann thermal heat bath. In this paper we present the solution of the FFPE in terms of n integral transformation. The transformation maps the solution of ordinary Fokker-Planck Equation onto the olution of the FFPE, and is based on Le vys generalized central limit theorem. The meaning of the transfor- mation is explained based on the known asymptotic solution of the continuous time random walk ~CTRW!.We nvestigate in detail ~i! a force-free particle, ~ii! a particle in a uniform field, and ~iii! a particle in a harmonic eld. We also find an exact solution of the CTRW, and compare the CTRW result with the corresponding olution of the FFPE. The relation between the fractional first passage time problem in an external nonlinear eld and the corresponding integer first passage time is given. An example of the one-dimensional fractional rst passage time in an external linear field is investigated in detail. The FFPE is shown to be compatible with he Scher-Montroll approach for dispersive transport, and thus is applicable in a large variety of disordered ystems. The simple FFPE approach can be used as a practical tool for a phenomenological description of ertain types of complicated transport phenomena.
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from continuous time random walks to the fractional Fokker Planck Equation
Physical Review E, 2000Co-Authors: Eli Barkai, Ralf Metzler, J KlafterAbstract:We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck Equation (FFPE). This Equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic Equation. For the force free case we compare between the CTRW solution and that of the FFPE.
Weihua Deng - One of the best experts on this subject based on the ideXlab platform.
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numerical algorithm for the time fractional Fokker Planck Equation
Journal of Computational Physics, 2007Co-Authors: Weihua DengAbstract:Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin Equations, or the generalized master Equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker-Planck Equation. In this paper, firstly the time fractional, the sense of Riemann-Liouville derivative, Fokker-Planck Equation is transformed into a time fractional ordinary differential Equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann-Liouville derivative and Caputo derivative. Then combining the predictor-corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error O(k^m^i^n^{^1^+^2^@a^,^2^})+O(h^2), and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for @a=1.0 with the ones of directly discretizing classical Fokker-Planck Equation, some numerical results for time fractional Fokker-Planck Equation with several different fractional orders are demonstrated and compared with each other, moreover for @a=0.8 the convergent order in space is confirmed and the numerical results with different time step sizes are shown.
P. Zhuang - One of the best experts on this subject based on the ideXlab platform.
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finite difference approximations for the fractional Fokker Planck Equation
Applied Mathematical Modelling, 2009Co-Authors: S Chen, P. ZhuangAbstract:The fractional Fokker–Planck Equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker–Planck Equation on a finite domain. The solvability, stability, consistency, and convergence of these methods are discussed. Their stability is proved by the energy method. Two numerical examples are also presented to evaluate these finite difference methods against the exact analytical solutions.