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

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.

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

Abstract and Applied Analysis ◽

10.1155/2013/324869 ◽

2013 ◽

Vol 2013 ◽

pp. 1-10 ◽

Cited By ~ 5

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.

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

Mathematics ◽

10.3390/math9182218 ◽

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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Approximate Solutions to a Cantilever Beam Using Optimal Homotopy Asymptotic Method

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.430.22 ◽

2013 ◽

Vol 430 ◽

pp. 22-26 ◽

Cited By ~ 1

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.

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

Open Physics ◽

10.1515/phys-2017-0072 ◽

2017 ◽

Vol 15 (1) ◽

pp. 620-626 ◽

Cited By ~ 1

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.

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Analytical approximate solutions to the Thomas-Fermi equation

Open Physics ◽

10.2478/s11534-014-0472-9 ◽

2014 ◽

Vol 12 (7) ◽

Cited By ~ 3

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

International Journal of Biomathematics ◽

10.1142/s1793524516500819 ◽

2016 ◽

Vol 09 (06) ◽

pp. 1650081 ◽

Cited By ~ 11

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.

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

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2014/682910 ◽

2014 ◽

Vol 2014 ◽

pp. 1-16 ◽

Cited By ~ 12

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.

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Application of Optimal Homotopy Asymptotic Method to Burger Equations

Journal of Applied Mathematics ◽

10.1155/2013/387478 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 11

Author(s):

R. Nawaz ◽

H. Ullah ◽

S. Islam ◽

M. Idrees

Keyword(s):

Asymptotic Method ◽

Decomposition Method ◽

Adomian Decomposition Method ◽

Approximate Solutions ◽

Adomian Decomposition ◽

Optimal Homotopy Asymptotic Method

We apply optimal homotopy asymptotic method (OHAM) for finding approximate solutions of the Burger's-Huxley and Burger's-Fisher equations. The results obtained by proposed method are compared to those of Adomian decomposition method (ADM) (Ismail et al., (2004)). As a result it is concluded that the method is explicit, effective, and simple to use.

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APPROXIMATE SOLUTION TO FRACTIONAL RICCATI DIFFERENTIAL EQUATIONS

Fractals ◽

10.1142/s0218348x19501287 ◽

2019 ◽

Vol 27 (08) ◽

pp. 1950128

Author(s):

MADIHA GOHAR

Keyword(s):

Differential Equation ◽

Close Agreement ◽

Approximate Solutions ◽

Random Processes ◽

Riccati Differential Equation ◽

Riccati Differential Equations ◽

Optimal Homotopy Asymptotic Method ◽

And Diffusion ◽

Efficiency And Reliability ◽

Diffusion Problems

In this paper, quadratic Riccati differential equation of fractional order has been solved by employing the optimal homotopy asymptotic method (Optimal HAM) with application to random processes, optimal control and diffusion problems. Optimal HAM uses simple computations with quite acceptable approximate solutions which have close agreement with exact solutions as compared to other techniques. To illustrate the efficiency and reliability of the method, some examples are provided and the results are discussed with tables.

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Stability Problems and Analytical Integration for the Clebsch’s System

Open Mathematics ◽

10.1515/math-2019-0018 ◽

2019 ◽

Vol 17 (1) ◽

pp. 242-259

Author(s):

Camelia Pop ◽

Remus-Daniel Ene

Keyword(s):

Numerical Integration ◽

Periodic Orbits ◽

Nonlinear Stability ◽

Asymptotic Method ◽

Approximate Solutions ◽

Equilibrium States ◽

Stability Problems ◽

Analytical Integration ◽

Optimal Homotopy Asymptotic Method

Abstract The nonlinear stability and the existence of periodic orbits of the equilibrium states of the Clebsch’s system are discussed.. Numerical integration using the Lie-Trotter integrator and the analytic approximate solutions using Multistage Optimal Homotopy Asymptotic Method are presented, too.

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