The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

Sakar, Mehmet Giyas; Erdoğan, Fevzi

doi:10.1016/j.apm.2013.03.074

Cited by 55 publications

(24 citation statements)

References 24 publications

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“…Keeping all this in mind, various type of vigorous techniques has been developed to find an approximate solution of such type of fractional differential equations, among others, generalized differential transform method [7], variational iteration method [8], local fractional variational iteration method [9], reproducing kernel Hilbert space method [10], Adomian decomposition method [11] and homotopy analysis method [12], Fractional reduced differential transform method [13,14,15,16].…”

Section: Introductionmentioning

confidence: 99%

Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay

Singh

Kumar

2017

SeMA

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Abstract. This paper deals the implementation of homotopy perturbation transform method (HPTM) for numerical computation of initial valued autonomous system of time-fractional partial differential equations (TFPDEs) with proportional delay, including generalized Burgers equations with proportional delay. The HPTM is a hybrid of Laplace transform and homotopy perturbation method. To confirm the efficiency and validity of the method, the computation of three test problems of TFPDEs with proportional delay presented. The proposed solutions are obtained in series form, converges very fast. The scheme seems very reliable, effective and efficient powerful technique for solving various type of physical models arising in sciences and engineering.

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Section: Introductionmentioning

confidence: 99%

Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay

Singh

Kumar

2017

SeMA

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“…Liao [24] was the first to propose the homotopy analysis method (HAM) which inhibits the use of small/large parameters and provide an easy and simplified technique to find the analytic solution of all kinds of linear and non-linear problems encountered in solid mechanics, fluid mechanics and so forth [25][26][27][28][29]. The method is provided with a tool which gives the opportunity to select and control the convergence region of the obtained solution.…”

Section: Analytical Approximations By Means Of Hammentioning

confidence: 99%

Analytic Approximate Solutions and Numerical Results for Stagnation Point Flow of Jeffrey Fluid Towards an Off-Centered Rotating Disk

Khan

Riaz

2014

J. mech.

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The present paper studies the three-dimensional, off centered stagnation flow of a Jeffrey fluid over a rotating disk. The governing non-linear equations and their associated boundary conditions are transformed into coupled ordinary differential equations by utilizing an appropriate similarity transformation. Homotopy analysis method is utilized to evaluate the analytical solution in the form of infinite series. Also, the convergence region of the obtained solution is determined and plotted. The effects of pertaining parameters on radial, azimuthal and induced velocities of the fluid flow are presented graphically and discussed. Moreover comparisons have also been made with the previous results as a special case.

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“…The author of [34] developed implicit unconditionally stable numerical methods to solve (1.1) on the condition that a.x/ D 0, b.x/ D 1 and f .x, t/ D 0. Sakar and Erdogan [35] introduced homotopy analysis method and Adomians decomposition method (ADM) for solving time-fractional Fornberg-Whitham equation. Also in [36], the approximate analytical solutions to the nonlinear Fornberg-Whitham equation with fractional time derivative has been obtained by using a reliable algorithm like the variational iteration method (VIM).…”

mentioning

confidence: 99%

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

Izadkhah

Saberi‐Nadjafi

2014

Math Methods in App Sciences

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In this paper, we study the numerical solution to time-fractional partial differential equations with variable coefficients that involve temporal Caputo derivative. A spectral method based on Gegenbauer polynomials is taken for approximating the solution of the given time-fractional partial differential equation in time and a collocation method in space. The suggested method reduces this type of equation to the solution of a linear algebraic system. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the proposed method. IntroductionRecently, fractional differential operators are indisputably found to play a fundamental role in the modeling of a considerable number of phenomena. Because of the nonlocal property of fractional derivative, they can utilize for modeling of memory-dependent phenomena and complex media such as porous media and anomalous diffusion [1][2][3][4]. They have been used in modeling turbulent flow [5,6], chaotic dynamics of classical conservation systems [7], and even finance [8,9] (see [1] for more information).Also, fractional calculus emerged as an important and efficient tool for the study of dynamical systems where classical methods reveal strong limitations. In [10], a fractional advection-dispersion equation is derived by extending Fick's first law from isotropic media to heterogeneous media and is particularly suitable for description of the highly skewed and heavy-tailed dispersion processes observed in rivers and other natural media.In the last decade or so, extensive research has been carried out on the development of numerical methods for fractional partial differential equations, including finite difference method [11][12][13], finite element methods [14,15], and spectral methods [16]. In [17][18][19], the authors used their proposed numerical schemes to solve Bagley-Torvik equation and other ordinary fractional differential equations. Gegenbauer polynomials have received much attention for their fundamental properties as well as for their use in applied mathematics. The described functions are a key ingredient to the implementation of spectral and pseudo-spectral methods to solve certain types of differential equations. Gegenbauer polynomials are a convenient basis for polynomial approximations because they are eigenfunctions of corresponding differential operators. For numerical methods, it is usually as convenient as efficient to convert between representations of a polynomial by expansion coefficients or by function values, respectively.Spectral approximations, such as the Fourier approximation based upon trigonometric polynomials for periodic problems, and the Chebyshev, Legendre, or the general Gegenbauer approximation based upon polynomials for nonperiodic problems are exponentially accurate for analytic functions [20][21][22][23]. In [24], the authors provided collocation method for natural convection heat transfer equations embedded in porous medium by using the rational Gegenbauer polynomials. Gottlieb and Shu in [25] used the Geg...

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The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method

Cited by 55 publications

References 24 publications

Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay

Homotopy perturbation transform method for solving fractional partial differential equations with proportional delay

Analytic Approximate Solutions and Numerical Results for Stagnation Point Flow of Jeffrey Fluid Towards an Off-Centered Rotating Disk

Gegenbauer spectral method for time‐fractional convection–diffusion equations with variable coefficients

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