scholarly journals Approximate Solutions for Flow with a Stretching Boundary due to Partial Slip

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
U. Filobello-Nino ◽  
H. Vazquez-Leal ◽  
A. Sarmiento-Reyes ◽  
B. Benhammouda ◽  
V. M. Jimenez-Fernandez ◽  
...  
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Perturbation Method ◽  
Nonlinear Differential Equation ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Partial Slip ◽  
Homotopy Perturbation

The homotopy perturbation method (HPM) is coupled with versions of Laplace-Padé and Padé methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to partial slip. Comparing results between approximate and numerical solutions, we concluded that our results are capable of providing an accurate solution and are extremely efficient.

scholarly journals Second Approximate Solution of Duffing Equation with Strong Nonlinearity by Homotopy Perturbation Method

1970 ◽  
Vol 30 ◽  
pp. 59-75
Author(s):  
M Alhaz Uddin ◽  
M Abdus Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Systems ◽  
Perturbation Method ◽  
Differential System ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Second Order ◽  
Strong Nonlinearity ◽  
Analytical Technique ◽  
Homotopy Perturbation

 In this paper, the second order approximate solution of a general second order nonlinear ordinary differential system, modeling damped oscillatory process is considered. The new analytical technique based on the work of He’s homotopy perturbation method is developed to find the periodic solution of a second order ordinary nonlinear differential system with damping effects. Usually the second or higher order approximate solutions are able to give better results than the first order approximate solutions. The results show that the analytical approximate solutions obtained by homotopy perturbation method are uniformly valid on the whole solutions domain and they are suitable not only for strongly nonlinear systems, but also for weakly nonlinear systems. Another advantage of this new analytical technique is that it also works for strongly damped, weakly damped and undamped systems. Figures are provided to show the comparison between the analytical and the numerical solutions. Keywords: Homotopy perturbation method; damped oscillation; nonlinear equation; strong nonlinearity. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 59-75  DOI: http://dx.doi.org/10.3329/ganit.v30i0.8504

Solution for Celebrated Blasius Problem Using Homotopy Perturbation Method

2020 ◽  
Vol 12 (2) ◽  
pp. 284-287
Author(s):  
Monika Rani ◽  
Vikramjeet Singh ◽  
Rakesh Goyal
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Perturbation Method ◽  
High Speed ◽  
Homotopy Perturbation Method ◽  
Initial Slope ◽  
Moving Wall ◽  
Homotopy Perturbation ◽  
Blasius Problem ◽  
Blasius Equation

In this manuscript, we have analyzed Celebrated Blasius boundary problem with moving wall or high speed 2D laminar viscous flow over gasifying flat plate. To find the way out of this nonlinear differential equation, a version of semi-analytical homotopy perturbation method has been applied. It has been observed that the precision of the solution would be achieved with increasing approximations. On comparison with literature, our solution has been proven highly accurate and valid with faster rate of convergence. It has been revealed that the second order approximate solution of Blasius equation in terms of initial slope is obtained as 0.33315 reducing the error by 0.32%.

scholarly journals On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

scholarly journals AN ACCURATE SOLUTION TO THE LOTKA-VOLTERRA EQUATIONS BY MODIFIED HOMOTOPY PERTURBATION METHOD

2012 ◽  
Vol 09 ◽  
pp. 326-333 ◽  
Author(s):  
M. S. H. CHOWDHURY ◽  
M. M. RAHMAN
Keyword(s):  
Perturbation Method ◽  
Numerical Solution ◽  
Nonlinear Equations ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Volterra Equations ◽  
Algebraic Equations ◽  
Time Intervals ◽  
Homotopy Perturbation

In this paper, we suggest a method to solve the multispecies Lotka-Voltera equations. The suggested method, which we call modified homotopy perturbation method, can be considered as an extension of the homotopy perturbation method (HPM) which is very efficient in solving a varety of differential and algebraic equations. The HPM is modified in order to obtain the approximate solutions of Lotka-Voltera equation response in a sequence of time intervals. In particular, the example of two species is considered. The accuracy of this method is examined by comparison with the numerical solution of the Runge-Kutta-Verner method. The results prove that the modified HPM is a powerful tool for the solution of nonlinear equations.

An easy and computable approximation for Troesch’s problem by using the Laplace Transform-Homotopy Perturbation Method

2019 ◽  
Vol 29 ◽  
pp. 1-14
Author(s):  
U. Filobello-Nino ◽  
H. Vazquez-Leal ◽  
A. L. Herrera-May ◽  
V. M. Jimenez-Fernandez ◽  
J. Cervantes-Perez ◽  
...  
Keyword(s):  
Approximate Solution ◽  
Laplace Transform ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Absolute Relative Error ◽  
Homotopy Perturbation ◽  
Average Absolute Relative Error ◽  
The Laplace Transform ◽  
Troesch’S Problem

This work introduces the Laplace Transform-Homotopy Perturbation Method (LT-HPM) in order to provide an approximate solution for Troesch’s problem. After comparing figures between exact and approximate solutions, as well as the average absolute relative error (AARE) of the approximate solutions of this research, with others reported in the literature, it can be said that the proposed solutions are accurate and handy. In conclusion, LT-HPM is a potentially useful tool.

The Homotopy Perturbation Method for Accurate Orbits of the Planets in the Solar System: The Elliptical Kepler Equation

2017 ◽  
Vol 72 (10) ◽  
pp. 933-940
Author(s):  
Aisha Alshaery
Keyword(s):  
Solar System ◽  
Perturbation Method ◽  
Series Solution ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Accurate Solution ◽  
Homotopy Perturbation ◽  
Halley's Comet ◽  
The Mean ◽  
Kepler Equation

AbstractAccurate trajectories for the orbits of the planets in our solar system depends on obtaining an accurate solution for the elliptical Kepler equation. This equation is solved in this article using the homotopy perturbation method. Several properties of the periodicity of the obtained approximate solutions are introduced through some lemmas. Numerically, our calculations demonstrated the applicability of the obtained approximate solutions for all the planets in the solar system and also in the whole domain of eccentricity and mean anomaly. In the whole domain of the mean anomaly, 0≤M≤2π, and by using the different approximate solutions, the residuals were less than 4×10−17 for e∈[0, 0.06], 4×10−9 for e∈[0.06, 0.25], 3×10−8 for e∈[0.25, 0.40], 3×10−7 for e∈[0.40, 0.50], and 10−6 for e∈[0.50, 1.0]. Also, the approximate solutions were compared with the Bessel–Fourier series solution in the literature. In addition, the approximate homotopy solutions for the eccentric anomaly are used to show the convergence and periodicity of the approximate radial distances of Mercury and Pluto for three and five periods, respectively, as confirmation for some given lemmas. It has also been shown that the present analysis can be successfully applied to the orbit of Halley’s comet with a significant eccentricity.

Orthogonal Flow Impinging on a Wall with Suction or Blowing

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Asmat Ara ◽  
Syed Anwer Ali ◽  
Muhammad Jamil
Keyword(s):  
Incompressible Fluid ◽  
Perturbation Method ◽  
Flat Plate ◽  
Skin Friction ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Viscous Incompressible Fluid ◽  
Approximate Solutions ◽  
Homotopy Perturbation ◽  
Porous Flat Plate

The goal of this work is the approximate solutions of a viscous incompressible fluid impinging orthogonally on a porous flat plate. The equation governing the flow of an incompressible fluid is investigated using the homotopy perturbation method (HPM) with the aid of Padé-approximants. The approximate solutions can be successfully applied to provide the value of the skin-friction. The reliability and efficiency of the approximate solutions were verified using numerical solutions in the literature.

Application of homotopy perturbation method to solve two models of delay differential equation systems

2017 ◽  
Vol 10 (06) ◽  
pp. 1750080 ◽  
Author(s):  
Şuayip Yüzbaşi ◽  
Murat Karaçayir
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Lorenz System ◽  
Approximate Solutions ◽  
Mathematical Physics ◽  
Differential Systems ◽  
Homotopy Perturbation ◽  
Delay Differential Systems ◽  
Delay Differential

In this paper, two delay differential systems are considered, namely, a famous model from mathematical biology about the spread of HIV viruses in blood and the advanced Lorenz system from mathematical physics. We then apply the homotopy perturbation method (HPM) to find their approximate solutions. It turns out that the method gives rise to easily obtainable solutions. In addition, residual error functions of the solutions are graphed and it is shown that increasing the parameter [Formula: see text] in the method improves the results in both cases.

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