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).

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 Solution of the Differential-Difference Equations by Optimal Homotopy Asymptotic Method

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
H. Ullah ◽  
S. Islam ◽  
M. Idrees ◽  
M. Fiza
Keyword(s):  
Difference Equations ◽  
Asymptotic Method ◽  
Volterra Equation ◽  
Approximate Solutions ◽  
Optimal Homotopy Asymptotic Method ◽  
Convergence Of Approximate Solutions ◽  
Efficiency And Reliability

We applied a new analytic approximate technique, optimal homotopy asymptotic method (OHAM), for treatment of differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Volterra equation in different form. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

scholarly journals Solution of Boundary Layer Problems with Heat Transfer by Optimal Homotopy Asymptotic Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
H. Ullah ◽  
S. Islam ◽  
M. Idrees ◽  
M. Arif
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Finite Difference ◽  
Asymptotic Method ◽  
Shooting Method ◽  
Approximate Solutions ◽  
Optimal Homotopy Asymptotic Method ◽  
Boundary Layer Problems ◽  
Convergence Of Approximate Solutions

Application of Optimal Homotopy Asymptotic Method (OHAM), a new analytic approximate technique for treatment of Falkner-Skan equations with heat transfer, has been applied in this work. To see the efficiency of the method, we consider Falkner-Skan equations with heat transfer. It provides us with a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature as finite difference (N. S. Asaithambi, 1997) and shooting method (Cebeci and Keller, 1971). The obtained solutions show that OHAM is effective, simpler, easier, and explicit.

scholarly journals Approximate and generalized solutions of conformable type Coudrey–Dodd–Gibbon–Sawada–Kotera equation

2020 ◽  
pp. 2150021
Author(s):  
Mehmet Senol ◽  
Lanre Akinyemi ◽  
Ayşe Ata ◽  
Olaniyi S. Iyiola
Keyword(s):  
Fractional Order ◽  
Approximate Solutions ◽  
Generalized Solutions ◽  
Analysis Method ◽  
Approximate Methods ◽  
Power Series Method ◽  
Closed Form Solutions ◽  
Homotopy Analysis ◽  
Residual Power Series ◽  
Residual Power

In this study, we consider conformable type Coudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation. Three powerful analytical methods are employed to obtain generalized solutions of the nonlinear equation of interest. First, the sub-equation method is used as baseline where generalized closed form solutions are obtained and are exact for any fractional order [Formula: see text]. Furthermore, residual power series method (RPSM) and [Formula: see text]-homotopy analysis method ([Formula: see text]-HAM) are then applied to obtain approximate solutions. These are possible using some properties of conformable derivative. These approximate methods are very powerful and efficient due to the absence of the need for linearization, discretization and perturbation. Numerical simulations are carried out showing error values, [Formula: see text]-curve for [Formula: see text]-HAM and the effects of fractional order on the solution profiles.

Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method

2013 ◽  
Vol 430 ◽  
pp. 22-26 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herisanu ◽  
Traian Marinca
Keyword(s):  
Small Parameter ◽  
Cantilever Beam ◽  
Asymptotic Method ◽  
Approximate Solutions ◽  
Lumped Mass ◽  
Engineering Problems ◽  
Approximate Frequency ◽  
Optimal Homotopy Asymptotic Method ◽  
Analytical Expressions

The response of a cantilever beam with a lumped mass attached to its free end subject to harmonical excitation at the base is investigated by means of the Optimal Homotopy Asymptotic Method (OHAM). Approximate accurate analytical expressions for the solutions and for approximate frequency are determined. This method does not require any small parameter in the equation. The obtained results prove that our method is very accurate, effective and simple for investigation of such engineering problems.

scholarly journals Constructing analytic solutions on the Tricomi equation

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 143-148 ◽  
Author(s):  
Emran Khoshrouye Ghiasi ◽  
Reza Saleh
Keyword(s):  
Homotopy Analysis Method ◽  
Series Solution ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Analysis Method ◽  
Auxiliary Parameter ◽  
Tricomi Equation ◽  
Homotopy Analysis ◽  
Optimal Values

AbstractIn this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

scholarly journals Application of the optimal homotopy asymptotic method to nonlinear Bingham fluid dampers

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 620-626 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Liviu Bereteu
Keyword(s):  
Asymptotic Method ◽  
Approximate Solutions ◽  
Bingham Fluid ◽  
Rapid Convergence ◽  
Bingham Model ◽  
Mr Dampers ◽  
Approximate Analytical Solutions ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Fluid Dampers

AbstractDynamic response time is an important feature for determining the performance of magnetorheological (MR) dampers in practical civil engineering applications. The objective of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to give approximate analytical solutions of the nonlinear differential equation of a modified Bingham model with non-viscous exponential damping. Our procedure does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. OHAM is very efficient in practice for ensuring very rapid convergence of the solution after only one iteration and with a small number of steps.

Analytical approximate solutions to the Thomas-Fermi equation

Open Physics ◽  
2014 ◽  
Vol 12 (7) ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene
Keyword(s):  
Excellent Agreement ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Thomas Fermi ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method

AbstractThe purpose of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to solve the nonlinear differential Thomas-Fermi equation. Our procedure does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. An excellent agreement was found between our approximate results and numerical solutions, which prove that OHAM is very efficient in practice, ensuring a very rapid convergence after only one iteration.

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