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.

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 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.

Application of Homotopy Perturbation Method with Chebyshev Polynomials to Nonlinear Problems

2010 ◽  
Vol 65 (1-2) ◽  
pp. 65-70
Author(s):  
Changbum Chun
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Chebyshev Polynomials ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Efficiency And Reliability

AbstractIn this paper, we present an efficient modification of the homotopy perturbation method by using Chebyshev’s polynomials and He’s polynomials to solve some nonlinear differential equations. Some illustrative examples are given to demonstrate the efficiency and reliability of the modified homotopy perturbation method.

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 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.

Rational Approximations for Heat Radiation and Troesch’s Equations

2016 ◽  
Vol 13 (03) ◽  
pp. 1650039 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Hüseyin Koçak ◽  
Inan Ates
Keyword(s):  
Differential Equation ◽  
Numerical Methods ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Heat Radiation ◽  
Rational Approximations ◽  
Rational Homotopy ◽  
Rational Solutions ◽  
Homotopy Perturbation

In this paper, a new tool for the solution of nonlinear differential equations is presented. It is named rational homotopy perturbation method (RHPM). It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. Furthermore, in order to deal with BVP problems, we propose a modification of RHPM method. The obtained results show that RHPM is a powerful tool capable to generate highly accurate rational solutions.

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.

scholarly journals Homotopy perturbation method for the solution of the electrostatic potential differential equation

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
Author(s):  
Li-Na Zhang ◽  
Ji-Huan He
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Strongly Nonlinear ◽  
Homotopy Perturbation ◽  
Explicit Analytical Solution ◽  
Approximate Analytical Solutions ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation

This paper obtains an explicit analytical solution for nonlinear Poisson-Boltzmann equation by the homotopy perturbation method, which does not require a small parameter in the equation under study, so it can be applied to both the weakly and strongly nonlinear problems. The obtained results show the evidence of the usefulness of the homotopy perturbation method for obtaining approximate analytical solutions for nonlinear equations.

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