scholarly journals Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states

2020 ◽  
Vol 146 (3) ◽  
pp. 481-511
Author(s):  
Daxin Nie ◽  
Jing Sun ◽  
Weihua Deng
Keyword(s):  
Numerical Algorithm ◽  
Space Time ◽  
Internal States ◽  
Fokker Planck

scholarly journals Analytical Solution of Space-Time Fractional Fokker-Planck Equation by Homotopy Perturbation Sumudu Transform Method

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ravi Shanker Dubey ◽  
Badr Saad T. Alkahtani ◽  
Abdon Atangana
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Space Time ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Sumudu Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

An efficient approach based on homotopy perturbation method by using Sumudu transform is proposed to solve some linear and nonlinear space-time fractional Fokker-Planck equations (FPEs) in closed form. The space and time fractional derivatives are considered in Caputo sense. The homotopy perturbation Sumudu transform method (HPSTM) is a combined form of Sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. Some examples show that the HPSTM is an effective tool for solving many space time fractional partial differential equations.

scholarly journals Numerical Solutions of Fractional Fokker-Planck Equations Using Iterative Laplace Transform Method

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Limei Yan
Keyword(s):  
Laplace Transform ◽  
Numerical Solutions ◽  
Convergent Series ◽  
Space Time ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Promising Tool ◽  
The Laplace Transform ◽  
Fokker Planck Equations ◽  
Fokker Planck

A relatively new iterative Laplace transform method, which combines two methods; the iterative method and the Laplace transform method, is applied to obtain the numerical solutions of fractional Fokker-Planck equations. The method gives numerical solutions in the form of convergent series with easily computable components, requiring no linearization or small perturbation. The numerical results show that the approach is easy to implement and straightforward when applied to space-time fractional Fokker-Planck equations. The method provides a promising tool for solving space-time fractional partial differential equations.

Effect of scale on solute dispersion in saturated porous media

1984 ◽  
Vol 16 (1) ◽  
pp. 18-18 ◽  
Author(s):  
Vijay K. Gupta ◽  
R. N. Bhatiacharya
Keyword(s):  
Differential Equation ◽  
Porous Media ◽  
Time Scale ◽  
Space Time ◽  
Saturated Porous Media ◽  
Solute Dispersion ◽  
Image Position ◽  
Steady Velocity ◽  
Fokker Planck

Consider a saturated porous m edium in which water is flowing slowly with a steady velocity. Suppose at some space-time scale the concentration C(x, r) of a non-reactive dilute solute is governed by the following Fokker-Planck differential equation:

scholarly journals Space-Time Inversion of Stochastic Dynamics

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 839
Author(s):  
Massimiliano Giona ◽  
Antonio Brasiello ◽  
Alessandra Adrover
Keyword(s):  
Stochastic Dynamics ◽  
Exit Time ◽  
Planck Equation ◽  
Space Time ◽  
Langevin Equations ◽  
Time Inversion ◽  
Stochastic Thermodynamics ◽  
Wiener Processes ◽  
Fokker Planck Equation ◽  
Fokker Planck

This article introduces the concept of space-time inversion of stochastic Langevin equations as a way of transforming the parametrization of the dynamics from time to a monotonically varying spatial coordinate. A typical physical problem in which this approach can be fruitfully used is the analysis of solute dispersion in long straight tubes (Taylor-Aris dispersion), where the time-parametrization of the dynamics is recast in that of the axial coordinate. This allows the connection between the analysis of the forward (in time) evolution of the process and that of its exit-time statistics. The derivation of the Fokker-Planck equation for the inverted dynamics requires attention: it can be deduced using a mollified approach of the Wiener perturbations “a-la Wong-Zakai” by considering a sequence of almost everywhere smooth stochastic processes (in the present case, Poisson-Kac processes), converging to the Wiener processes in some limit (the Kac limit). The mathematical interpretation of the resulting Fokker-Planck equation can be obtained by introducing a new way of considering the stochastic integrals over the increments of a Wiener process, referred to as stochastic Stjelties integrals of mixed order. Several examples ranging from stochastic thermodynamics and fractal-time models are also analyzed.

Polynomials for numerical solutions of space-time fractional differential equations (of the Fokker–Planck type)

2019 ◽  
Vol 36 (9) ◽  
pp. 2996-3015
Author(s):  
Jiao Wang
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Space Time ◽  
Algebraic Equations ◽  
Operator Matrices ◽  
Content Type ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Fokker Planck

Purpose Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type. Design/methodology/approach The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations. Findings Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique. Originality/value Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

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