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.

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.

scholarly journals Analytic Approximate Solution for Falkner-Skan Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Rapid Convergence ◽  
Boundary Layer Problem ◽  
Small Parameters ◽  
Optimal Homotopy Asymptotic Method ◽  
Good Agreement ◽  
Layer Problem

This paper deals with the Falkner-Skan nonlinear differential equation. An analytic approximate technique, namely, optimal homotopy asymptotic method (OHAM), is employed to propose a procedure to solve a boundary-layer problem. Our method does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. The obtained results reveal that this procedure is very effective, simple, and accurate. 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.

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.

scholarly journals Different Approximations to the Solution of Upper-Convected Maxwell Fluid over a Porous Stretching Plate

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Vasile Marinca ◽  
Remus-Daniel Ene ◽  
Bogdan Marinca ◽  
Romeo Negrea
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Maxwell Fluid ◽  
Approximate Solutions ◽  
Two Dimensional ◽  
Rapid Convergence ◽  
Similarity Transformations ◽  
Optimal Homotopy Asymptotic Method ◽  
Stretching Plate ◽  
Good Agreement

In the present paper, we consider an incompressible magnetohydrodynamic flow of two-dimensional upper-convected Maxwell fluid over a porous stretching plate with suction and injection. The nonlinear partial differential equations are reduced to an ordinary differential equation by the similarity transformations and taking into account the boundary layer approximations. This equation is solved approximately by means of the optimal homotopy asymptotic method (OHAM). This approach is highly efficient and it controls the convergence of the approximate solutions. Different approximations to the solution are given, showing the exceptionally good agreement between the analytical and numerical solutions of the nonlinear problem. OHAM is very efficient in practice, ensuring a very rapid convergence of the solutions after only one iteration even though it does not need small or large parameters in the governing equation.

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 Optimal homotopy asymptotic method to large post-buckling deformation of MEMS

2018 ◽  
Vol 148 ◽  
pp. 13003 ◽  
Author(s):  
Nicolae Herisanu ◽  
Vasile Marinca
Keyword(s):  
Differential Equation ◽  
Numerical Solution ◽  
Approximate Method ◽  
Excellent Agreement ◽  
Electrostatic Field ◽  
Asymptotic Method ◽  
Integro Differential Equation ◽  
Analytical Results ◽  
Post Buckling ◽  
Optimal Homotopy Asymptotic Method

In the present paper, the post-buckling response of an axially stressed clamped-clamped actuator, modeled as a beam and subjected to a symmetric electrostatic field is analyzed. An analytical approximate method, namely the Optimal Homotopy Asymptotic Method (OHAM) is applied to the governing nonlinear integro-differential equation. The analytical results obtained through the proposed procedure show excellent agreement with numerical solution, proving the validity of the proposed procedure, which is simple and easy to use.

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 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 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.

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