scholarly journals Application of Local Fractional Homotopy Perturbation Method in Physical Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Nabard Habibi ◽  
Zohre Nouri
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Models ◽  
Applied Mathematics ◽  
Nonlinear Phenomena ◽  
Homotopy Perturbation ◽  
Physical Problems ◽  
Fourier Series Method ◽  
Physical Phenomena ◽  
Good Agreement

Nonlinear phenomena have important effects on applied mathematics, physics, and issues related to engineering. Most physical phenomena are modeled according to partial differential equations. It is difficult for nonlinear models to obtain the closed form of the solution, and in many cases, only an approximation of the real solution can be obtained. The perturbation method is a wave equation solution using HPM compared with the Fourier series method, and both methods results are good agreement. The percentage of error of ux,t with α=1 and 0.33, t =0.1 sec, between the present research and Yong-Ju Yang study for x≥0.6 is less than 10. Also, the % error for x≥0.5 in α=1 and 0.33, t =0.3 sec, is less than 5, whereas for α=1 and 0.33, t =0.8 and 0.7 sec, the % error for x≥0.4 is less than 8.

Numerical inversion method for the Laplace transform based on Boubaker polynomials operational matrix

2021 ◽  
Author(s):  
Rachid Belgacem ◽  
Ahmed Bokhari ◽  
Salih Djilali ◽  
Sunil Kumar
Keyword(s):  
Laplace Transform ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Operational Matrix ◽  
Inversion Method ◽  
Numerical Inversion ◽  
Homotopy Perturbation ◽  
Boubaker Polynomials ◽  
The Laplace Transform ◽  
Good Agreement

We investigate through this research the numerical inversion technique for the Laplace transforms cooperated by the integration Boubaker polynomials operational matrix. The efficiency of the presented approach is demonstrated by solving some differential equations. Also, this technique is combined with the standard Laplace Homotopy Perturbation Method. The numerical results highlight that there is a very good agreement between the estimated solutions with exact solutions.

scholarly journals Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes ◽  
Yasir Khan ◽  
Uriel Filobello-Nino ◽  
Alejandro Diaz-Sanchez
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Homotopy Perturbation ◽  
Physical Phenomena

The fact that most of the physical phenomena are modelled by nonlinear differential equations underlines the importance of having reliable methods for solving them. This work presents the rational biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose the Laplace-Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is expressed as the quotient of two truncated power series. The usage of the method is illustrated with two case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions.

scholarly journals Homotopy perturbation method for a Stefan problem with variable latent heat

2014 ◽  
Vol 18 (2) ◽  
pp. 391-398 ◽  
Author(s):  
R Rajeev
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Perturbation Method ◽  
Latent Heat ◽  
Stefan Problem ◽  
Homotopy Perturbation Method ◽  
Homotopy Perturbation ◽  
Variable Latent Heat ◽  
Good Agreement

In this paper, homotopy perturbation method is successfully applied to find an approximate solution of one phase Stefan problem with variable latent heat. The results thus obtained are compared graphically with a published analytical solution and are in good agreement.

scholarly journals Solving the Fractional Rosenau-Hyman Equation via Variational Iteration Method and Homotopy Perturbation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
R. Yulita Molliq ◽  
M. S. M. Noorani
Keyword(s):  
Perturbation Method ◽  
Iteration Method ◽  
Homotopy Perturbation Method ◽  
Applied Mathematics ◽  
Analytical Techniques ◽  
Variational Iteration Method ◽  
Convergent Series ◽  
Alternative Methods ◽  
Variational Iteration ◽  
Homotopy Perturbation

In this study, fractional Rosenau-Hynam equations is considered. We implement relatively new analytical techniques, the variational iteration method and the homotopy perturbation method, for solving this equation. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for fractional Rosenau-Hynam equations. In these schemes, the solution takes the form of a convergent series with easily computable components. The present methods perform extremely well in terms of efficiency and simplicity.

Investigation of Casimir and Van der Waals Forces for a Nonlinear Double-Clamped Beam Using Homotopy Perturbation Method

2009 ◽  
Author(s):  
Mahdi Mojahedi ◽  
Hamid Moeenfard ◽  
Mohammad Taghi Ahmadian
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Axial Loading ◽  
Design Parameters ◽  
Electric Force ◽  
Homotopy Perturbation ◽  
Static Responses ◽  
Fringing Field ◽  
Fringing Field Effect ◽  
Good Agreement

In this study, static deflection and Instability of double-clamped nanobeams actuated by electrostatic field and intermolecular force, are investigated. The model accounts for the electric force nonlinearity of the excitation and for the fringing field effect. Effects of mid-plane stretching and axial loading are considered. Galerkin’s decomposition method is utilized to convert the nonlinear differential equation of motion to a nonlinear algebraic equation which is solved using the homotopy perturbation method. The effect of the design parameters such as axial load and mid-plane stretching on the static responses and pull-in instability is discussed. Results are in good agreement with presented in the literature.

scholarly journals A Simple Modification of Homotopy Perturbation Method for the Solution of Blasius Equation in Semi-Infinite Domains

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
M. Aghakhani ◽  
M. Suhatril ◽  
M. Mohammadhassani ◽  
M. Daie ◽  
A. Toghroli
Keyword(s):  
Boundary Condition ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Simple Modification ◽  
Infinite Domains ◽  
Homotopy Perturbation ◽  
Blasius Equation ◽  
Fifth Order ◽  
Adomian’S Polynomials ◽  
Good Agreement

A simple modification of the homotopy perturbation method is proposed for the solution of the Blasius equation with two different boundary conditions. Padé approximate is used to deal with the boundary condition at infinity. The results obtained from the analytical method are compared to Howarth’s numerical solution and fifth order Runge-Kutta Fehlberg method indicating a very good agreement. The proposed method is a simple and reliable modification of homotopy perturbation method, which does not require the existence of a small parameter, linearization of the equation, or computation of Adomian’s polynomials.

scholarly journals Analytical Solution of Liénard Differential Equation using Homotopy Perturbation Method

2019 ◽  
Vol 39 ◽  
pp. 87-100
Author(s):  
Md Mamun Ur Rashid Khan ◽  
Goutam Saha
Keyword(s):  
Differential Equation ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Research Work ◽  
Approximate Solutions ◽  
Homotopy Perturbation ◽  
Actual Solution ◽  
Different Types ◽  
Taylor’S Series ◽  
Good Agreement

In this research work, the well-known Homotopy perturbation method (HPM) is used to find the approximate solutions of the nonlinear Liénard differential equation (LDE) using different types of boundary conditions. In order to find the accuracy of the approximate solution, one term, two terms and three terms HPM approximations are considered. This idea is actually based on the idea of Taylor’s series polynomials. It is found that solution converges to the actual solution with the increase of the terms in the guess solution. Moreover, in each of the new HPM solution, previously obtained solutions are added to it in order to find the exactness of HPM solutions. However, the nature of the solution seems to be complicated. In addition, comparisons are made with the previously published results and a good agreement is observed. GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 87-100

Homotopy perturbation method for predicting tsunami wave propagation with crisp and uncertain parameters

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Perumandla Karunakar ◽  
Snehashish Chakraverty
Keyword(s):  
Wave Propagation ◽  
Perturbation Method ◽  
Water Wave ◽  
Design Methodology ◽  
Homotopy Perturbation Method ◽  
Tsunami Wave ◽  
Uncertain Parameters ◽  
Content Type ◽  
Homotopy Perturbation ◽  
Good Agreement

Purpose The purpose of this paper is to find the solution of classical nonlinear shallow-water wave (SWW) equations in particular to the tsunami wave propagation in crisp and interval environment. Design/methodology/approach Homotopy perturbation method (HPM) has been used for handling crisp and uncertain differential equations governing SWW equations. Findings The wave height and depth-averaged velocity of a tsunami wave in crisp and interval cases have been obtained. Originality/value Present results by HPM are compared with the existing solution (in crisp case), and they are found to be in good agreement. Also, the residual error of the solutions is found for the convergence conformation and reliability of the present results.

scholarly journals Rational Homotopy Perturbation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Héctor Vázquez-Leal
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Homotopy Perturbation ◽  
Solution Methods ◽  
Rational Version ◽  
Physical Phenomena

The solution methods of nonlinear differential equations are very important because most of the physical phenomena are modelled by using such kind of equations. Therefore, this work presents a rational version of homotopy perturbation method (RHPM) as a novel tool with high potential to find approximate solutions for nonlinear differential equations. We present two case studies; for the first example, a comparison between the proposed method and the HPM method is presented; it will show how the RHPM generates highly accurate approximate solutions requiring less iteration, in comparison to results obtained by the HPM method. For the second example, which is a Van der Pol oscillator problem, we compare RHPM, HPM, and VIM, finding out that RHPM method generates the most accurate approximated solution.

scholarly journals Solution of the Davey–Stewardson equation using homotopuy analysis method

2010 ◽  
Vol 15 (4) ◽  
pp. 423-433 ◽  
Author(s):  
H. Jafari ◽  
M. Alipour
Keyword(s):  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Analysis Method ◽  
Homotopy Perturbation Method ◽  
Analysis Method ◽  
Nls Equation ◽  
Homotopy Perturbation ◽  
Higher Dimensional ◽  
Homotopy Analysis ◽  
Good Agreement

In this paper, the homotopy analysis method (HAM) proposed by Liao is adopted for solving Davey–Stewartson (DS) equations which arise as higher dimensional generalizations of the nonlinear Schrödinger (NLS) equation. The results obtained by HAM have been compared with the exact solutions and homotopy perturbation method (HPM) to show the accuracy of the method. Comparisons indicate that there is a very good agreement between the HAM solutions and the exact solutions in terms of accuracy.

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