An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method

2016 ◽  
Vol 10 ◽  
pp. 1131-1150 ◽  
Author(s):  
Mohammad Hamarsheh ◽  
Ahmad Izani Ismail ◽  
Zaid Odibat
Keyword(s):  
Fractional Order ◽  
Asymptotic Method ◽  
Analytic Solution ◽  
Riccati Equations ◽  
Optimal Homotopy Asymptotic Method

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 Optimal Homotopy Asymptotic Method for Solving Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
N. Ratib Anakira ◽  
A. K. Alomari ◽  
I. Hashim
Keyword(s):  
Differential Equations ◽  
Asymptotic Method ◽  
Delay Differential Equations ◽  
Perturbation Methods ◽  
Analytic Solution ◽  
Approximate Analytic Solution ◽  
Initial Value ◽  
Optimal Homotopy Asymptotic Method ◽  
Delay Differential ◽  
First Time

We extend for the first time the applicability of the optimal homotopy asymptotic method (OHAM) to find the algorithm of approximate analytic solution of delay differential equations (DDEs). The analytical solutions for various examples of linear and nonlinear and system of initial value problems of DDEs are obtained successfully by this method. However, this approach does not depend on small or large parameters in comparison to other perturbation methods. This method provides us with a convenient way to control the convergence of approximation series. The results which are obtained revealed that the proposed method is explicit, effective, and easy to use.

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 The Optimal Homotopy Asymptotic Method for Solving Two Strongly Fractional-Order Nonlinear Benchmark Oscillatory Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2218
Author(s):  
Mohd Taib Shatnawi ◽  
Adel Ouannas ◽  
Ghenaiet Bahia ◽  
Iqbal M. Batiha ◽  
Giuseppe Grassi
Keyword(s):  
Fractional Order ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Auxiliary Function ◽  
Approximate Methods ◽  
Homotopy Analysis ◽  
Optimal Homotopy Asymptotic Method ◽  
Relativistic Oscillator ◽  
Convergence Of Approximate Solutions

This paper proceeds from the perspective that most strongly nonlinear oscillators of fractional-order do not enjoy exact analytical solutions. Undoubtedly, this is a good enough reason to employ one of the major recent approximate methods, namely an Optimal Homotopy Asymptotic Method (OHAM), to offer approximate analytic solutions for two strongly fractional-order nonlinear benchmark oscillatory problems, namely: the fractional-order Duffing-relativistic oscillator and the fractional-order stretched elastic wire oscillator (with a mass attached to its midpoint). In this work, a further modification has been proposed for such method and then carried out through establishing an optimal auxiliary linear operator, an auxiliary function, and an auxiliary control parameter. In view of the two aforesaid applications, it has been demonstrated that the OHAM is a reliable approach for controlling the convergence of approximate solutions and, hence, it is an effective tool for dealing with such problems. This assertion is completely confirmed by performing several graphical comparisons between the OHAM and the Homotopy Analysis Method (HAM).

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