An analytic algorithm for the space–time fractional advection–dispersion equation

Pandey, Ram K.; Singh, Om Prakash; Baranwal, Vipul K.

doi:10.1016/j.cpc.2011.01.015

Cited by 71 publications

(31 citation statements)

References 45 publications

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“…Many analytical algorithms have been investigated analytically to obtain closed form solutions in treating FDEs such as variational iteration method, Fourier transform method, the homotopy analysis method, the method of separation of variables, Adomian decomposition method, and Laplace transform method [10,23,[27][28][29]. However, there are only a few types of these equations in which the analytical solutions are available.…”

Section: Introductionmentioning

confidence: 99%

An efficient fractional polynomial method for space fractional diffusion equations

Krishnaveni

Kannan

Balachandar

et al. 2018

Ain Shams Engineering Journal

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In this paper, we develop a new approximation technique for solving space fractional diffusion equation. The method of solution is based on fractional order Legendre function with the concept of Caputo's definition of fractional derivatives. Convergence analysis and error bound of the method are discussed. Several Illustrative examples are included to demonstrate the validity and applicability of the proposed method. The obtained results reveal that the method is more accurate and efficient than the methods such as Chebyshev finite difference method and Tau approach method discussed in the literature. Ó 2016 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Section: Introductionmentioning

confidence: 99%

An efficient fractional polynomial method for space fractional diffusion equations

Krishnaveni

Kannan

Balachandar

et al. 2018

Ain Shams Engineering Journal

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“…Various definitions and basic concept of fractional calculus are present in many books [1][2][3][4]. Therefore, several analytical and numerical methods were developed for solutions of fractional differential equations (both linear and nonlinear), among which Adomian's decomposition method [5][6][7], variation iteration method [8,9], homotopy perturbation method [10][11][12], homotopy analysis method [13][14][15], homotopy asymptotic method [16][17][18],differential transform method [19], and Galerkin method [20].…”

Section: Introductionmentioning

confidence: 99%

A Modified Analytical Approach for Fractional Discrete KdV Equations Arising in Particle Vibrations

Kumar

2017

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.

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In this paper, we present an effectiveness of the proposed analytical algorithm for nonlinear time fractional discrete KdV equations. The discrete KdV equation play an important role in modeling of complicated physical phenomena such as particle vibrations in lattices, current flow in electrical flow and pulse in biological chains. The proposed algorithm is amagalation of the homotopy analysis method, Laplace transform method and homotopy polynomials. First, we present an alternative framework of the proposed method which can be used simply and effectively to handle nonlinear problems arising in science and engineering. Then, mainly, we address the convergence study of nonlinear fractional differential equations with Laplace transform. Camparisons are made between the results of the proposed method and exact solutions. Illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method. The new results reveal that the proposed method is explicit, efficient and easy to use.

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“…We can also mention the implicit difference method based on the shifted Grünwald-Letnikov approximation [36], transformation of fractional differential equation into a system of ordinary differential equation [37], the random walk algorithms [38,39], the spectral regularization method [52], the Crank-Nicholson difference scheme [53], Adomian's decomposition [50], a spatial and temporal discretization [64], the fractional variational iteration method [54], the homotopy perturbation method [51,63,72] and the Jacobi collocation method [73].…”

Section: Introductionmentioning

confidence: 99%

Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation

Povstenko

Kyrylych

2017

Entropy

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Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically.

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An analytic algorithm for the space–time fractional advection–dispersion equation

Cited by 71 publications

References 45 publications

An efficient fractional polynomial method for space fractional diffusion equations

An efficient fractional polynomial method for space fractional diffusion equations

A Modified Analytical Approach for Fractional Discrete KdV Equations Arising in Particle Vibrations

Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation

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