Application of He’s Homotopy Perturbation Method for Solving Fractional Fokker-Planck Equations

2009 ◽  
Vol 64 (12) ◽  
pp. 788-794 ◽  
Author(s):  
Mohamed M. Mousa ◽  
Aidarkhan Kaltayev
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Planck Equation ◽  
Approximate Solutions ◽  
Convergent Series ◽  
Homotopy Perturbation ◽  
External Force Field ◽  
Promising Tool ◽  
Transport Problems ◽  
Fokker Planck

Abstract The fractional Fokker-Planck equation (FFPE) has been used in many physical transport problems which take place under the influence of an external force field and other important applications in various areas of engineering and physics. In this paper, by means of the homotopy perturbation method (HPM), exact and approximate solutions are obtained for two classes of the FFPE initial value problems. The method gives an analytic solution in the form of a convergent series with easily computed components. The obtained results show that the HPM is easy to implement, accurate and reliable for solving FFPEs. The method introduces a promising tool for solving other types of differential equation with fractional order derivatives

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 Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
M. S. H. Chowdhury ◽  
I. Hashim ◽  
S. Momani ◽  
M. M. Rahman
Keyword(s):  
Numerical Methods ◽  
Perturbation Method ◽  
Fourth Order ◽  
Chaotic Systems ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Homotopy Perturbation ◽  
Promising Tool ◽  
Dynamical Behaviors

Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy-perturbation method (MHPM) is applied to the Chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy-perturbation method (HPM). The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth-order Runge-Kutta (RK4) solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

EXACT SOLITARY-WAVE SOLUTIONS FOR THE NONLINEAR DISPERSIVE K(2,2,1) and K(3,3,1) EQUATIONS BY THE HOMOTOPY PERTURBATION METHOD

2009 ◽  
Vol 23 (30) ◽  
pp. 3667-3675 ◽  
Author(s):  
AHMET YILDIRIM
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Convergent Series ◽  
Perturbation Series ◽  
Solitary Wave Solutions ◽  
Homotopy Perturbation ◽  
Wave Solutions ◽  
Explicit Exact Solution

We implemented homotopy perturbation method for approximating the solution to the nonlinear dispersive K(m,n,1) type equations. By using this scheme, the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. To illustrate the application of this method, numerical results are derived by using the calculated components of the homotopy perturbation series.

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