Beyond Adomian method: The variational iteration method for solving heat-like and wave-like equations with variable coefficients

Shou, Dahua; He, Ji Huan

doi:10.1016/j.physleta.2007.07.011

Cited by 63 publications

(42 citation statements)

References 30 publications

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“…Substituting (15), (20) and (33) into (10) gives the following equation for x 2 (t): = e 1 cos ωt + e 3 cos 3ωt + . .…”

Section: Formulation and Solution Approachmentioning

confidence: 99%

Approximate Solutions for Conservative Nonlinear Oscillators by He’s Homotopy Method

Yebra¹

2008

Zeitschrift Für Naturforschung A

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He's homotopy perturbation method is adapted to calculate higher-order approximate periodic solutions of conservative nonlinear oscillators for which the elastic force term is an odd nonlinear function. The 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. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions. An example is presented to illustrate the usefulness and effectiveness of the proposed technique, and comparison of the results obtained using this method with those obtained using other techniques reveals that the former is very effective and convenient.

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“…Substituting (15), (20) and (33) into (10) gives the following equation for x 2 (t): = e 1 cos ωt + e 3 cos 3ωt + . .…”

Section: Formulation and Solution Approachmentioning

confidence: 99%

Approximate Solutions for Conservative Nonlinear Oscillators by He’s Homotopy Method

Yebra¹

2008

Zeitschrift Für Naturforschung A

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“…6 Recently, there has been a growing interest for obtaining the explicit solutions to heat-like and wave-like models by analytic techniques. The different types of heat-like and wave-like physical models in physics have been solved by Wazwaz and Gorguis, 7 Shou and He, 8 and Ö zis and Agırseven. 9 In this article, the homotopy analysis transform method (HATM) basically illustrates how the Laplace transform can be used to approximate analytical solutions of the wave-like fractional models by manipulating the homotopy analysis method (HAM).…”

Section: Introductionmentioning

confidence: 99%

A modified homotopy analysis method for solution of fractional wave equations

Yin

Kumar

2015

Advances in Mechanical Engineering

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The aim of this article is to introduce a modified analytical approach to obtain quick and accurate solution of wave-like fractional physical models. This modified analytical approach is an innovative adjustment in Laplace transform algorithm and homotopy analysis method for fractional partial differential equations. The proposed technique solves the problems using Adomian's polynomials. The homotopy analysis transform method utilizes a simple and powerful method to adjust and control the convergence region of the infinite series solution using an auxiliary parameter. The numerical solutions obtained by this modified proposed method indicate that the approach is easy to implement, highly accurate, and computationally very attractive. A good agreement between the obtained solutions and some well-known results has been obtained.

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“…According to the energy balance method [2], locating at some a special point, that is, xt = p/4 and setting R(t = p/4x) = 0, we can obtain an approximate frequencyamplitude relationship of the studied nonlinear oscillator. Such treatment is much simple and has been widely used by engineers [3][4][5][6][7]. The accuracy of such location method, however, strongly depends upon the chosen location point.…”

Section: The Description Of Hamiltonian Approach Methodsmentioning

confidence: 99%

“…To overcome the shortcomings, many techniques have appeared in open literature, for example, parameterexpanding method [1][2][3], variational iteration method [4][5][6][7], energy balance method [8][9][10][11][12][13][14], variational approach method [15][16][17][18][19], amplitude-frequency formulation [20][21][22], homotopy perturbation method [23][24][25][26][27][28][29], and the other analytical approximate solutions [30].…”

Section: Introductionmentioning

confidence: 99%

Analytical study of nonlinear oscillatory systems using the Hamiltonian approach technique

2014

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In this article, we investigate and apply Hamiltonian approach method as one of the analytical approximate techniques, for studying the strongly nonlinear dynamical systems such as the motion of a rigid rod rocking back on the circular surface without slipping and the free vibrations of an autonomous conservative oscillator with inertia and static-type fifth-order nonlinearities. To illustrate the applicability and accuracy of the method, the approximate solution results are compared with exact and numerical solutions.

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Beyond Adomian method: The variational iteration method for solving heat-like and wave-like equations with variable coefficients

Cited by 63 publications

References 30 publications

Approximate Solutions for Conservative Nonlinear Oscillators by He’s Homotopy Method

Approximate Solutions for Conservative Nonlinear Oscillators by He’s Homotopy Method

A modified homotopy analysis method for solution of fractional wave equations

Analytical study of nonlinear oscillatory systems using the Hamiltonian approach technique

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