Homotopy Perturbation Pade Technique for Solving Fractional Riccati Differential Equations

Jafari, Hossein; Tajadodi, H.; Matikolai, S. A. Hosseini

doi:10.1515/ijnsns.2010.11.s1.271

Cited by 26 publications

(21 citation statements)

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“…But equation (12) for one term approximation of series respect to p and for yields (18) From equations (18) and (17) we can easily find that the solution w is (19) Replacing w from equation (19) into equation (11) yields:…”

Section: Equation Of Motionmentioning

confidence: 99%

“…There have been several classical approaches employed to solve the governing nonlinear differential equations to study the nonlinear vibrations including perturbation methods [18], He's Max-Min Approach (MMA) [2], He's Energy Balance Method [19], Combined Homotopy Variational Approach [20], Iteration perturbation method [21], Homotopy perturbation method (HPM) [22,23], Multistage Adomian Decomposition Method [24], Variational iteration method [3], Multiple scales method [25], Monotone iteration schemes [26], ADM-Padé technique [27], Navier and Levy-type solution [28], Hamiltonian approach [29], Parameter Perturbation Method [30], Differential Transform method [31], Laplace Transform method [32] . The application of new equivalent function for deadzone and preload nonlinearities on the dynamical behavior of beam vibration using PEM has been investigated by [6][7][8].…”

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confidence: 99%

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High precise analysis of lateral vibration of quintic nonlinear beam

Sedighi

Reza

2013

Lat. Am. j. solids struct.

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This article intends to achieve a new formulation of beam vibration with quintic nonlinearity, including exact expressions for the beam curvature. To attain a proper design of the beam structures, it is essential to realize how the beam vibrates in its transverse mode which in turn yields the natural frequency of the system. In this direction, new powerful analytical method called Parameter Expansion Method (PEM) is employed to obtain the exact solution of frequency-amplitude relationship. Afterwards, it is clearly shown that the first term in series expansions is sufficient to produce a highly accurate approximation of mentioned system. Finally, preciseness of the present analytic procedures is evaluated in contrast with numerical calculations methods, giving excellent results. Nonlinear vibration of beams is of substantial interest for engineers and has been studied, considerably. From the engineering outlook, structures like helicopter rotor blades, space craft antennae, flexible satellites, airplane wings, robot arms, high-rise buildings, long-span bridges, drill strings and vibratory drilling can be modelled as a beam-like member. The problem of the transversely vibrating beam was recently formulated in terms of the partial differential equation of motion by many researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] with different boundary conditions. Sedighi et al. [1] presented the advantages of recent modern analytical approaches applied on the governing equation of transversely vibrating cantilever beams. A comprehensive study on the analytical investigation of cubic nonlinear vibrating tapered beams has been conducted by Bayat et al. [2]. The analytical expression for geometrically non-linear vibration of clamped-clamped Euler-Bernoulli beams including cubic non-linear strain-displacement relationship has been obtained by Barari et al. [3]. The application of new analytical approaches on the dynamical behavior of beams vibration with different boundary conditions has been investigated by Sedighi et al. [5][6][7][8]. Closed form solution to free vibration of beams with mixed boundary conditions has been proposed by Motaghian et al. [9]. Nikkhah Bahrami et al.[10] used modified wave approach for calculation of natural frequencies and mode shapes of arbitrary non-uniform beams. Non-linear modal analysis of a rotating beams studied by [11,12]. When the vibration amplitudes are moderate or large, the geometric nonlinearity must be included.It is very important to provide an accurate analysis towards the understanding of the non-linear vibration characteristics of these structures. Most models dealing with nonlinear dynamics of flexible

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Section: Equation Of Motionmentioning

confidence: 99%

mentioning

confidence: 99%

High precise analysis of lateral vibration of quintic nonlinear beam

Sedighi

Reza

2013

Lat. Am. j. solids struct.

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“…Recently, Sev-eral methods have been used to solve fractional differential equations. These include the Laplace transform method [3], Fourier transform method [4], Adomians decomposition method [7], Homotopy analysis method [8,9], Homotopy perturbation method [10,11], Variational iteration method [12][13][14][15][16][17], Legendre wavelets [18] B-spline collocation method [19], collocation method [20] and so on. In this paper, we present a numerical technique to solve fractional differential equation…”

Section: Introductionmentioning

confidence: 99%

Numerical solution of fractional differential equations by using fractional B-spline

et al. 2013

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Abstract:In this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work. PACS

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“…Numerical and analytical methods have included the finite difference method [7][8][9], Adomian decomposition method [10][11][12][13][14], variational iteration method [15][16][17][18], homotopy perturbation method [19][20][21][22], generalized differential transform method [23][24][25][26], homotopy analysis method [27,28], and other methods [1,29].…”

Section: Introductionmentioning

confidence: 99%

Application of Legendre wavelets for solving fractional differential equations

Jafari

Yousefi

Firoozjaee

et al. 2011

Computers & Mathematics with Applications

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168

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a b s t r a c tIn this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.

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Homotopy Perturbation Pade Technique for Solving Fractional Riccati Differential Equations

Cited by 26 publications

References 1 publication

High precise analysis of lateral vibration of quintic nonlinear beam

High precise analysis of lateral vibration of quintic nonlinear beam

Numerical solution of fractional differential equations by using fractional B-spline

Application of Legendre wavelets for solving fractional differential equations

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