An efficient analytical approach for solving fractional Fokker-Planck equations (original) (raw)
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2015
The fractional Fokker-Plank equation paid much more attention by researchers due to its wide range of applications in several areas of sciences and engineering. In this article, our main motto is to present a new approximate solution of time-fractional FPE by means of a new semi- analytical approach called fractional reduced differential transform method (FRDTM), with appropriate initial condition. In FRDTM, fractional derivative is considered in the Caputo sense. The validity and efficiency of FRDTM is illustrated by considering three numeric experiments. The solutions behavior and the effects of different values of fractional order are depicted
A Comparative Analysis of Fractional-Order Fokker–Planck Equation
Symmetry
The importance of partial differential equations in physics, mathematics and engineering cannot be emphasized enough. Partial differential equations are used to represent physical processes, which are then solved analytically or numerically to examine the dynamical behaviour of the system. The new iterative approach and the Homotopy perturbation method are used in this article to solve the fractional order Fokker–Planck equation numerically. The Caputo sense is used to characterize the fractional derivatives. The suggested approach’s accuracy and applicability are demonstrated using illustrations. The proposed method’s accuracy is expressed in terms of absolute error. The proposed methods are found to be in good agreement with the exact solution of the problems using graphs and tables. The results acquired using the given approaches are also obtained at various fractional orders of the derivative. It is observed from the graphs and tables that fractional order solutions converge to ...
Toward computational algorithm for time-fractional Fokker–Planck models
Advances in Mechanical Engineering
This article describes an efficient algorithm based on residual power series to approximate the solution of a class of partial differential equations of time-fractional Fokker–Planck model. The fractional derivative is assumed in the Caputo sense. The proposed algorithm gives the solution in a form of rapidly convergent fractional power series with easily computable coefficients. It does not require linearization, discretization, or small perturbation. To test simplicity, potentiality, and practical usefulness of the proposed algorithm, illustrative examples are provided. The approximate solutions of time-fractional Fokker–Planck equations are obtained by the residual power series method are compared with those obtained by other existing methods. The present results and graphics reveal the ability of residual power series method to deal with a wide range of partial fractional differential equations emerging in the modeling of physical phenomena of science and engineering.
An Analytical Study on Fractional Fokker-Planck Equation by Homotopy Analysis Transform Method
Recent Advances in Mathematics, Statistics and Computer Science, 2016
In this article, we introduce an analytical method, namely Homotopy Analysis Transform Method (HATM) which is a combination of Homotopy Analysis Method (HAM) and Laplace Decomposition Method (LDM). This scheme is simple to apply linear and nonlinear fractional differential equations and having less computational work in comparison of other exiting methods. The most useful advantage of this method is to solve the fractional nonlinear differential equation without using Adomian polynomials and He's polynomials for the computation of nonlinear terms. The proposed method has no linearization and restrictive assumptions throughout the process. Presently, homotopy analysis transform method is used to solve time fractional Fokker-Planck equation and similar equations. The series solution obtained by HATM converges very fast. A good agreement between the obtained solution and some well-known results has been demonstrated.
Fractional Fokker-Planck Equation
Mathematics, 2017
We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain.
Numerical solution of the space fractional Fokker–Planck equation
Journal of Computational and Applied Mathematics, 2004
The traditional second-order Fokker-Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of α-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker-Planck equation except that the order α of the highest derivative is fractional.
Advances in Difference Equations
This paper provides a numerical approach for solving the time-fractional Fokker-Planck equation (FFPE). The authors use the shifted Chebyshev collocation method and the finite difference method (FDM) to present the fractional Fokker-Planck equation into systems of nonlinear equations; the Newton-Raphson method is used to produce approximate results for the nonlinear systems. The results obtained from the FFPE demonstrate the simplicity and efficiency of the proposed method.