scholarly journals Dynamical Study of Fokker-Planck Equations by Using Optimal Homotopy Asymptotic Method

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 264 ◽  
Author(s):  
H. Younas ◽  
Muhammad Mustahsan ◽  
Tareq Manzoor ◽  
Nadeem Salamat ◽  
S. Iqbal
Keyword(s):  
Convergence Rate ◽  
Exact Solutions ◽  
Fractional Order ◽  
Asymptotic Method ◽  
Dynamical Study ◽  
Optimal Homotopy Asymptotic Method ◽  
Relative Errors ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this article, Optimal Homotopy Asymptotic Method (OHAM) is used to approximate results of time-fractional order Fokker-Planck equations. In this work, 3rd order results obtained through OHAM are compared with the exact solutions. It was observed that results from OHAM have better convergence rate for time-fractional order Fokker-Planck equations. The solutions are plotted and the relative errors are tabulated.

scholarly journals Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
S. Zuhra ◽  
S. Islam ◽  
M. Idrees ◽  
Rashid Nawaz ◽  
I. A. Shah ◽  
...  
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Boundary Value ◽  
Singular Boundary ◽  
Point Boundary ◽  
Singular Boundary Value Problems ◽  
Optimal Homotopy Asymptotic Method

In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.

Mathematical study of fractional-order biological population model using optimal homotopy asymptotic method

2016 ◽  
Vol 09 (06) ◽  
pp. 1650081 ◽  
Author(s):  
S. Sarwar ◽  
M. A. Zahid ◽  
S. Iqbal
Keyword(s):  
Fractional Order ◽  
Asymptotic Method ◽  
Population Model ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Population Models ◽  
Mathematical Study ◽  
Third Order ◽  
Biological Population ◽  
Optimal Homotopy Asymptotic Method

In this paper, we study the fractional-order biological population models (FBPMs) with Malthusian, Verhulst, and porous media laws. The fractional derivative is defined in Caputo sense. The optimal homotopy asymptotic method (OHAM) for partial differential equations (PDEs) is extended and successfully implemented to solve FBPMs. Third-order approximate solutions are obtained and compared with the exact solutions. The numerical results unveil that the proposed extension in the OHAM for fractional-order differential problems is very effective and simple in computation. The results reveal the effectiveness with high accuracy and extremely efficient to handle most complicated biological population models.

scholarly journals Applications of OHAM and MOHAM for Fractional Seventh-Order SKI Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Jafar Biazar ◽  
Saghi Safaei
Keyword(s):  
Approximate Solution ◽  
Comparative Study ◽  
Exact Solutions ◽  
Asymptotic Method ◽  
Fractional Derivatives ◽  
Fractional Partial Differential Equations ◽  
Large Interval ◽  
Seventh Order ◽  
Optimal Homotopy Asymptotic Method ◽  
Ito Equation

In this article, a comparative study between optimal homotopy asymptotic method and multistage optimal homotopy asymptotic method is presented. These methods will be applied to obtain an approximate solution to the seventh-order Sawada-Kotera Ito equation. The results of optimal homotopy asymptotic method are compared with those of multistage optimal homotopy asymptotic method as well as with the exact solutions. The multistage optimal homotopy asymptotic method relies on optimal homotopy asymptotic method to obtain an analytic approximate solution. It actually applies optimal homotopy asymptotic method in each subinterval, and we show that it achieves better results than optimal homotopy asymptotic method over a large interval; this is one of the advantages of this method that can be used for long intervals and leads to more accurate results. As far as the authors are aware that multistage optimal homotopy asymptotic method has not been yet used to solve fractional partial differential equations of high order, we have shown that this method can be used to solve these problems. The convergence of the method is also addressed. The fractional derivatives are described in the Caputo sense.

Random Vibrations in Discrete Nonlinear Dynamic Systems

1968 ◽  
Vol 10 (2) ◽  
pp. 168-174 ◽  
Author(s):  
R. G. Bhandari ◽  
R. E. Sherrer
Keyword(s):  
Exact Solutions ◽  
Density Function ◽  
Gaussian White Noise ◽  
Hermite Polynomials ◽  
Planck Equation ◽  
Degree Of Freedom ◽  
Multiple Series ◽  
Special Cases ◽  
Fokker Planck Equations ◽  
Fokker Planck

A one-degree-of-freedom system and a two-degree-of-freedom system containing Dis-placement and velocity dependent nonlinearities subjected to stationary gaussian white noise excitation have been studied by the method of the Fokker-Planck equation. Non-linearities have been represented by suitable polynomials. The Fokker-Planck equations governing the stationary probability density function for these systems have been solved by representing the density function by a multiple series of Hermite polynomials. The constants in the series expansion were determined by Galerkin's method. Analysis is developed for the system containing nonlinearities described by suitable polynomials in velocity and displacement dependent forces. Comparisons were made between series and exact solutions for those special cases for which exact solutions are known.

Exact solutions of Fokker-Planck equations associated to quantum wave functions

1998 ◽  
Vol 245 (1-2) ◽  
pp. 1-10 ◽  
Author(s):  
Nicola Cufaro Petroni ◽  
Salvatore De Martino ◽  
Silvio De Siena
Keyword(s):  
Exact Solutions ◽  
Wave Functions ◽  
Quantum Wave ◽  
Fokker Planck Equations ◽  
Fokker Planck

scholarly journals On the Solutions of Fractional Burgers-Fisher and Generalized Fisher’s Equations Using Two Reliable Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
A. K. Gupta ◽  
S. Saha Ray
Keyword(s):  
Exact Solutions ◽  
Asymptotic Method ◽  
Arbitrary Order ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Haar Wavelet ◽  
Wavelet Method ◽  
Haar Wavelet Method ◽  
Optimal Homotopy Asymptotic Method ◽  
Reliable Methods

Two reliable techniques, Haar wavelet method and optimal homotopy asymptotic method (OHAM), are presented. Haar wavelet method is an efficient numerical method for the numerical solution of arbitrary order partial differential equations like Burgers-Fisher and generalized Fisher equations. The approximate solutions thus obtained for the fractional Burgers-Fisher and generalized Fisher equations are compared with the optimal homotopy asymptotic method as well as with the exact solutions. Comparison between the obtained solutions with the exact solutions exhibits that both the featured methods are effective and efficient in solving nonlinear problems. The obtained results justify the applicability of the proposed methods for fractional order Burgers-Fisher and generalized Fisher’s equations.

scholarly journals On a class of exact solutions to the Fokker–Planck equations

1982 ◽  
Vol 23 (6) ◽  
pp. 1155-1158 ◽  
Author(s):  
L. Garrido ◽  
J. Masoliver
Keyword(s):  
Exact Solutions ◽  
Fokker Planck Equations ◽  
Fokker Planck
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