On the polynomial solution of the first Painlevé equation

2017 ◽  
Vol 6 (1) ◽  
pp. 34 ◽  
Author(s):  
D. Sierra Porta ◽  
L. A. Núñez
Keyword(s):  
Asymptotic Method ◽  
Numerical Solutions ◽  
Applied Mathematics ◽  
Polynomial Solution ◽  
Painlevé Equation ◽  
Theoretical Physics ◽  
Approximation Technique ◽  
Polynomial Solutions ◽  
Optimal Homotopy Asymptotic Method ◽  
Painleve Equation

The Painlevé equations and their solutions arises in pure, applied mathematics and theoretical physics. In this manuscript we apply the Optimal Homotopy Asymptotic Method (OHAM) for solving the first Painlevé equation. Our approximation technique is based on the use of polynomial solutions, which are shown to be accurate when compared to the computed numerical solutions, thus providing a very close description of the evolution of the system.

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.

Use of Optimal Homotopy Asymptotic Method and Galerkin’s Finite Element Formulation in the Study of Heat Transfer Flow of a Third Grade Fluid Between Parallel Plates

2011 ◽  
Vol 133 (9) ◽  
Author(s):  
S. Iqbal ◽  
A. R. Ansari ◽  
A. M. Siddiqui ◽  
A. Javed
Keyword(s):  
Heat Transfer ◽  
Finite Element ◽  
Poiseuille Flow ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Parallel Plates ◽  
Third Grade Fluid ◽  
Grade Fluid ◽  
Optimal Homotopy Asymptotic Method ◽  
Transfer Flow

We investigate the effectiveness of the optimal homotopy asymptotic method (OHAM) in solving nonlinear systems of differential equations. In particular we consider the heat transfer flow of a third grade fluid between two heated parallel plates separated by a finite distance. The method is successfully applied to study the constant viscosity models, namely plane Couette flow, plane Poiseuille flow, and plane Couette–Poiseuille flow for velocity fields and the temperature distributions. Numerical solutions of the systems are also obtained using a finite element method (FEM). A comparative analysis between the semianalytical solutions of OHAM and numerical solutions by FEM are presented. The semianalytical results are found to be in good agreement with numerical solutions. The results reveal that the OHAM is precise, effective, and easy to use for such systems of nonlinear differential equations.

scholarly journals An Algorithm: Optimal Homotopy Asymptotic Method for Solutions of Systems of Second-Order Boundary Value Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Rafiq Mufti ◽  
Muhammad Imran Qureshi ◽  
Salem Alkhalaf ◽  
S. Iqbal
Keyword(s):  
Exact Solutions ◽  
Boundary Value Problems ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Small Perturbations ◽  
Forcing Function ◽  
Optimal Homotopy Asymptotic Method ◽  
Linear And Nonlinear Systems

Optimal homotopy asymptotic method (OHAM) is proposed to solve linear and nonlinear systems of second-order boundary value problems. OHAM yields exact solutions in just single iteration depending upon the choice of selecting some part of or complete forcing function. Otherwise, it delivers numerical solutions in excellent agreement with exact solutions. Moreover, this procedure does not entail any discretization, linearization, or small perturbations and therefore reduces the computations a lot. Some examples are presented to establish the strength and applicability of this method. The results reveal that the method is very effective, straightforward, and simple to handle systems of boundary value problems.

scholarly journals Dual Approximate Solutions of the Unsteady Viscous Flow over a Shrinking Cylinder with Optimal Homotopy Asymptotic Method

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene
Keyword(s):  
Viscous Flow ◽  
Asymptotic Method ◽  
Numerical Solutions ◽  
Similarity Transformation ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Unsteady Viscous Flow ◽  
Optimal Homotopy Asymptotic Method ◽  
Shrinking Surface ◽  
Good Agreement

The unsteady viscous flow over a continuously shrinking surface with mass suction is investigated using the optimal homotopy asymptotic method (OHAM). The nonlinear differential equation is obtained by means of the similarity transformation. The dual solutions exist for a certain range of mass suction and unsteadiness parameters. A very good 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.

NUMERICAL SOLUTIONS OF TELEGRAPH EQUATIONS USING AN OPTIMAL HOMOTOPY ASYMPTOTIC METHOD

2015 ◽  
Vol 46 (8) ◽  
pp. 699-712
Author(s):  
Shaukat Iqbal ◽  
Muhammad Sadiq Hashmi ◽  
Nargis Khan ◽  
M. Ramzan ◽  
Amir H. Dar
Keyword(s):  
Asymptotic Method ◽  
Numerical Solutions ◽  
Telegraph Equations ◽  
Optimal Homotopy Asymptotic Method
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