Solution of an Integro-Differential Equation Arising in Oscillating Magnetic Fields Using He's Homotopy Perturbation Method

Dehghan, Mehdi; Shakeri, Fatemeh

doi:10.2528/pier07090403

Cited by 205 publications

(114 citation statements)

References 41 publications

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“…The homotopy perturbation method [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38], which requires neither a small parameter nor a linear term in a differential equation, yields a very rapid convergence of the solution series; in most cases only one iteration leads to high accuracy of the solution.…”

Section: Introductionmentioning

confidence: 99%

Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

Beléndez¹,

Pascual²,

Fernández³

et al. 2008

Phys. Scr.

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A modified He's homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He's homotopy perturbation method is modified by truncating the infinite series corresponding to the first order approximate solution before introducing this solution in the second order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulas obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.

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Section: Introductionmentioning

confidence: 99%

Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

Beléndez¹,

Pascual²,

Fernández³

et al. 2008

Phys. Scr.

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show abstract

“…The structure of method includes a deformation from intial solution to the final step that is obtained our favorite solution. This method has been used by many researchers to solve different types of non linear problems [9,10,14,21,22,23,34,35,12]. HPM is considered as a combination of the classical perturbation technique and the homotopy from topology in pure mathematics It is not restricted to small parameters like traditional perturbation methods.We can find highly accurate solutions in just few iterations.…”

Section: Structure Of Homotopy Perturbation Method(hpm)mentioning

confidence: 99%

Optimal Homotopy Asymptotic and Homotopy Perturbation Methods for Linear Mixed Volterra-Fredholm Integral Equations

Kasmaei

Rashidinia

2016

Nevşehir Bilim Ve Teknoloji Dergisi

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“…The homotopy perturbation method is also very often used for seeking the solution of integral equations of different kind [25][26][27][28][29][30][31][32][33][34][35][36][37][38]. Convergence of this method in case of integral equations is considered only in few papers.…”

Section: Introductionmentioning

confidence: 99%

An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit

Hetmaniok¹,

Słota²,

Trawiński³

et al. 2014

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In this paper an application of the homotopy perturbation method for solving the general linear integral equations of the second kind is discussed. It is shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in the homotopy perturbation method is convergent. The error of approximate solution, received by taking only the partial sum of the series, is also estimated. Moreover, there is presented an example of applying the method for approximate solution of an equation which has a practical application for charge calculation in supply circuit of the flash lamps used in cameras.

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Solution of an Integro-Differential Equation Arising in Oscillating Magnetic Fields Using He's Homotopy Perturbation Method

Cited by 205 publications

References 41 publications

Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

Optimal Homotopy Asymptotic and Homotopy Perturbation Methods for Linear Mixed Volterra-Fredholm Integral Equations

An analytical technique for solving general linear integral equations of the second kind and its application in analysis of flash lamp control circuit

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