scholarly journals Improvement of the homotopy perturbation method to nonlinear problems

2018 ◽  
Vol 13 (4) ◽  
pp. 43-53
Author(s):  
F. Alzaidi J. ◽  
A. Alderremy A.
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Homotopy Perturbation

ADDENDUM: NEW INTERPRETATION OF HOMOTOPY PERTURBATION METHOD

2006 ◽  
Vol 20 (18) ◽  
pp. 2561-2568 ◽  
Author(s):  
JI-HUAN HE
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Problems ◽  
Proper Understanding ◽  
Homotopy Perturbation ◽  
New Interpretation ◽  
Guided Tour

The present work constitutes a guided tour through the mathematics needed for a proper understanding of homotopy perturbation method as applied to various nonlinear problems. It gives a new interpretation of the concept of constant expansion in the homotopy perturbation method.

scholarly journals Solving Heat and Wave-Like Equations Using He's Polynomials

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Syed Tauseef Mohyud-Din
Keyword(s):  
Perturbation Method ◽  
Decomposition Method ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Problems ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Clear Advantage ◽  
Suggested Technique ◽  
Adomian’S Polynomials

We use He's polynomials which are calculated form homotopy perturbation method (HPM) for solving heat and wave-like equations. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. Several examples are given to verify the reliability and efficiency of the method. The fact that suggested technique solves nonlinear problems without using Adomian's polynomials is a clear advantage of this algorithm over the decomposition method.

scholarly journals A NEW NUMERICAL METHOD TO SOLVE NONLINEAR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 26 (3) ◽  
pp. 469-478
Author(s):  
Jinjiao Hou ◽  
Jing Niu ◽  
Welreach Ngolo
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Nonlinear Problems ◽  
Reproducing Kernel Method ◽  
Homotopy Perturbation ◽  
Linear Problems

In this paper, a new method combining the simplified reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-differential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the effectiveness of the approach.

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 The Application of the Homotopy Perturbation Method and the Homotopy Analysis Method to the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Hassan A. Zedan ◽  
Eman El Adrous
Keyword(s):  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Analysis Method ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Nonlinear Problems ◽  
Analysis Method ◽  
Homotopy Perturbation ◽  
Homotopy Analysis ◽  
The Absolute

We introduce two powerful methods to solve the generalized Zakharov equations; one is the homotopy perturbation method and the other is the homotopy analysis method. The homotopy perturbation method is proposed for solving the generalized Zakharov equations. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions; the homotopy analysis method is applied to solve the generalized Zakharov equations. HAM is a strong and easy-to-use analytic tool for nonlinear problems. Computation of the absolute errors between the exact solutions of the GZE equations and the approximate solutions, comparison of the HPM results with those of Adomian’s decomposition method and the HAM results, and computation the absolute errors between the exact solutions of the GZE equations with the HPM solutions and HAM solutions are presented.

A heuristic review on the homotopy perturbation method for non-conservative oscillators

2021 ◽  
pp. 146134842110592
Author(s):  
Chun-Hui He ◽  
Yusry O El-Dib
Keyword(s):  
Exact Solution ◽  
Perturbation Method ◽  
Analytical Methods ◽  
Homotopy Perturbation Method ◽  
Duffing Oscillator ◽  
Nonlinear Problems ◽  
Nonlinear Oscillators ◽  
Complex Problem ◽  
Review Article ◽  
Homotopy Perturbation

The homotopy perturbation method (HPM) was proposed by Ji-Huan. He was a rising star in analytical methods, and all traditional analytical methods had abdicated their crowns. It is straightforward and effective for many nonlinear problems; it deforms a complex problem into a linear system; however, it is still developing quickly. The method is difficult to deal with non-conservative oscillators, though many modifications have appeared. This review article features its last achievement in the nonlinear vibration theory with an emphasis on coupled damping nonlinear oscillators and includes the following categories: (1) Some fallacies in the study of non-conservative issues; (2) non-conservative Duffing oscillator with three expansions; (3)the non-conservative oscillators through the modified homotopy expansion; (4) the HPM for fractional non-conservative oscillators; (5) the homotopy perturbation method for delay non-conservative oscillators; and (6) quasi-exact solution based on He’s frequency formula. Each category is heuristically explained by examples, which can be used as paradigms for other applications. The emphasis of this article is put mainly on Ji-Huan He’s ideas and the present authors’ previous work on the HPM, so the citation might not be exhaustive.

The Variational Iteration Method and the Variational Homotopy Perturbation Method for Solving the KdV-Burgers Equation and the Sharma-Tasso-Olver Equation

2010 ◽  
Vol 65 (1-2) ◽  
pp. 25-33 ◽  
Author(s):  
Elsayed M. E. Zayed ◽  
Hanan M. Abdel Rahman
Keyword(s):  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Alternative Methods ◽  
Variational Iteration ◽  
Homotopy Perturbation

AbstractIn this article, two powerful analytical methods called the variational iteration method (VIM) and the variational homotopy perturbation method (VHPM) are introduced to obtain the exact and the numerical solutions of the (2+1)-dimensional Korteweg-de Vries-Burgers (KdVB) equation and the (1+1)-dimensional Sharma-Tasso-Olver equation. The main objective of the present article is to propose alternative methods of solutions, which avoid linearization and physical unrealistic assumptions. The results show that these methods are very efficient, convenient and can be applied to a large class of nonlinear problems.

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