Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition

Li, Xicheng; Xu, Mingyu; Jiang, Xiaoyun

doi:10.1016/j.amc.2008.12.023

Cited by 70 publications

(42 citation statements)

References 21 publications

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“…But these fractional differential equations are difficult to get their exact solutions [9,12,13,21]. So, these types of equations are solved by various methods such as Adomian decomposition method [1,22], variational iteration method [7,20], differential transform method [2,6], homotopy perturbation method [11,15], an iterative method [4,23], finite element method [19,25], finite difference method [3,18], etc..…”

Section: Introductionmentioning

confidence: 99%

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Li¹,

Sun²,

Yin³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, we develop a new application of the Mittag-Leffler function that will extend the application to fractional homogeneous differential equations, and propose a Mittag-Leffler function undetermined coefficient method. A new solution is constructed in power series. When a very simple ordinary differential equation is satisfied, no matter the original equation is linear or nonlinear, the method is valid, then combine the alike terms, compare the coefficient with identical powers, and the undetermined coefficient will be obtained. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided, and the solutions are in the form of generalized Mittag-Leffler function. The results reveal that the approach introduced here are very effective and convenient for solving homogeneous differential equations with fractional order.

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Section: Introductionmentioning

confidence: 99%

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Li¹,

Sun²,

Yin³

et al. 2017

J. Nonlinear Sci. Appl.

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show abstract

“…Nowadays, integer-order differential equations have been extended to the fractional-order equations, which have been applied in almost every field of physics [1][2][3], finance [4,5], hydrology [6], engineering [7], mathematics [8] and science. These new equations are more adequate than the previously used integer-order equations.…”

Section: Introductionmentioning

confidence: 99%

Finite difference methods for fractional dispersion equations

Wang

2010

Applied Mathematics and Computation

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“…In another recent article of Yao [12], it is seen that the fractal geometry theory is combined with seepage flow mechanism to establish the nonlinear diffusion equation of fluid flow in fractal reservoir. It is to be noted that some works on fractional diffusion equations have already been done by Li et al [13], Ganji and Sadighi [14] etc., using various mathematical techniques. But to the best of authors' knowledge the convergence of the solution of the considered nonlinear fractional problem by the minimization of residual error has not yet been studied by any researcher.…”

Section: Introductionmentioning

confidence: 99%

Solution of the Nonlinear Fractional Diffusion Equation with Absorbent Term and External Force Using Optimal Homotopy-Analysis Method

Vishal¹,

Das²

2012

Zeitschrift Für Naturforschung A

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In this article, the optimal homotopy-analysis method (HAM) is used to obtain approximate analytic solutions of the time-fractional nonlinear diffusion equation in the presence of an external force and an absorbent term. The fractional derivatives are considered in the Caputo sense to avoid nonzero derivative of constants. Unlike usual HAM this method contains at the most three convergence control parameters which determine the fast convergence of the solution through different choices of convergence control parameters. Effects of proper choice of parameters on the convergence of the approximate series solution by minimizing the averaged residual error for different particular cases are depicted through tables and graphs.

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Homotopy perturbation method to time-fractional diffusion equation with a moving boundary condition

Cited by 70 publications

References 21 publications

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

A new Mittag-Leffler function undetermined coefficient method and its applications to fractional homogeneous partial differential equations

Finite difference methods for fractional dispersion equations

Solution of the Nonlinear Fractional Diffusion Equation with Absorbent Term and External Force Using Optimal Homotopy-Analysis Method

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