scholarly journals Spectral-Homotopy Perturbation Method for Solving Governing MHD Jeffery-Hamel Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Nonlinear Boundary Value Problems ◽  
Pseudospectral Methods ◽  
New Approach ◽  
Homotopy Perturbation ◽  
Chebyshev Pseudospectral

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the MHD Jeffery-Hamel flow and the effect of MHD on the flow has been discussed. Comparisons are made between the proposed technique, the previous studies, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the presented approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method at small orders. The MATLAB software has been used to solve all the equations in this study.

scholarly journals A New Spectral-Homotopy Perturbation Method and Its Application to Jeffery-Hamel Nanofluid Flow with High Magnetic Field

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Magnetic Field ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Nanofluid Flow ◽  
New Approach ◽  
Homotopy Perturbation ◽  
Chebyshev Pseudospectral

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method, and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the Jeffery-Hamel flow considering the effects of magnetic field and nanoparticle. Comparisons are made between the proposed technique, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the present approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method.

scholarly journals Numerical Solutions of Fourth-Order Fractional Integro-Differential Equations

2010 ◽  
Vol 65 (12) ◽  
pp. 1027-1032 ◽  
Author(s):  
Ahmet Yıldırım ◽  
Sefa Anıl Sezer ◽  
Yasemin Kaplan
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Fourth Order ◽  
Fractional Derivatives ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Methods Numerical

In this paper, the homotopy perturbation method (HPM) is developed to obtain numerical solutions of linear and nonlinear boundary value problems for fourth-order fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. Some examples are given and comparisons are made; the comparisons show that the homotopy perturbation method is very effective and convenient and overcome the difficulty of traditional methods. Numerical examples are presented to illustrate the efficiency, simplicity, and reliability of the method.

On Spectral-Homotopy Perturbation Method Solution of Nonlinear Differential Equations in Bounded Domains

2013 ◽  
Vol 1 (1) ◽  
pp. 25-37
Author(s):  
Ahmed A. Khidir
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Iteration Algorithm ◽  
Test Problems ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Spectral Technique ◽  
Homotopy Analysis

In this study, a combination of the hybrid Chebyshev spectral technique and the homotopy perturbation method is used to construct an iteration algorithm for solving nonlinear boundary value problems. Test problems are solved in order to demonstrate the efficiency, accuracy and reliability of the new technique and comparisons are made between the obtained results and exact solutions. The results demonstrate that the new spectral homotopy perturbation method is more efficient and converges faster than the standard homotopy analysis method. The methodology presented in the work is useful for solving the BVPs consisting of more than one differential equation in bounded domains. 

scholarly journals Homotopy Perturbation Method to Obtain Positive Solutions of Nonlinear Boundary Value Problems of Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hossein Jafari ◽  
Khadijeh Bagherian ◽  
Seithuti P. Moshokoa
Keyword(s):  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Adomian Decomposition Method ◽  
Type Problem ◽  
Nonlinear Boundary Value Problems ◽  
Homotopy Perturbation ◽  
Adomian Decomposition

We use the homotopy perturbation method for solving the fractional nonlinear two-point boundary value problem. The obtained results by the homotopy perturbation method are then compared with the Adomian decomposition method. We solve the fractional Bratu-type problem as an illustrative example.

scholarly journals Solution of nonlinear problems by a new analytical technique using Daftardar-Gejji and Jafari polynomials

2019 ◽  
Vol 11 (12) ◽  
pp. 168781401989696 ◽  
Author(s):  
Liaqat Ali ◽  
Saeed Islam ◽  
Taza Gul ◽  
Iraj Sadegh Amiri
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Boundary ◽  
Nonlinear Term ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Nonlinear Boundary Value Problems ◽  
Analytical Technique ◽  
Homotopy Perturbation

This article shows the solution of nonlinear differential equations by a new analytical technique called modified optimal homotopy perturbation method. Daftardar-Gejji and Jafari polynomials are used in the proposed method for the expansion of nonlinear term in the equation. Four nonlinear boundary value problems of fourth, fifth, sixth, and eighth orders are solved by modified optimal homotopy perturbation method as well as optimal homotopy perturbation method. The achieved consequences are authenticated by comparison with the results gained by the existing method—optimal homotopy perturbation method. The method consists of few steps and gives better results. The easy applicability and fast convergence are goals of the applied technique. The applied technique has fewer limitations and can be used for the phenomena containing ordinary differential equation, partial differential equation, integro-differential equation, and their systems.

A New Modification of The HPM for The Duffing Equation With High Nonlinearity

2013 ◽  
Vol 2 (2) ◽  
pp. 124-133
Author(s):  
Ahmed Khdir
Keyword(s):  
Perturbation Method ◽  
Spectral Methods ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
High Accuracy ◽  
Duffing Equation ◽  
Nonlinear Ordinary Differential Equations ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Pseudo Spectral

In this work we introduce a new modification of the homotopy perturbation method for solving nonlinear ordinary differential equations. The technique is based on the blending of the Chebyshev pseudo-spectral methods and the homotopy perturbation method (HPM). The method is tested by solving the strongly nonlinear Duffing equation for undamped oscillators. Comparison is made between the proposed technique, the standard HPM, an earlier modification of the HPM and the numerical solutions to demonstrate the high accuracy, applicability and validity of the present approach.

Application of Homotopy Perturbation Method for Harmonically Forced Duffing Systems

2011 ◽  
Vol 110-116 ◽  
pp. 2277-2283 ◽  
Author(s):  
Xiang Meng Zhang ◽  
Ben Li Wang ◽  
Xian Ren Kong ◽  
A Yang Xiao
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Primary Resonance ◽  
Effective Technique ◽  
Duffing System ◽  
First Order ◽  
Homotopy Perturbation ◽  
Analytical Approximations ◽  
Case Based

In this paper, He’s homotopy perturbation method (HPM) is applied to solve harmonically forced Duffing systems. Non-resonance of an undamped Duffing system and the primary resonance of a damped Duffing system are studied. In the former case, the first-order analytical approximations to the system’s natural frequency and periodic solution are derived by HPM, which agree well with the numerical solutions. In the latter case, based on HPM, the first-order approximate solution and the frequency-amplitude curves of the system are acquired. The results reveal that HPM is an effective technique to the forced Duffing systems.

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