On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Hikal, M.M.; Ibrahim, Mohamed A.

doi:10.12732/ijpam.v104i1.4

Cited by 3 publications

(6 citation statements)

References 24 publications

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“…where b:c denotes the ceiling function. This is the same as obtained by Hikal and Abu Ibrahim [12] using ADM.…”

Section: Advances In Mathematical Physicssupporting

confidence: 77%

“…To solve the space-time fractional ADE, Momani and Odibat [7] utilized the ADM and variational iteration approach. In this continuation, Yildirim and Koçak [11] solve the space-time fractional ADE by applying homotopy perturbation technique in Caputo sense and Hikal and Abu Ibrahim [12] solved it by the Adomian decomposition method. Alliche and Chikh [13] studied the nonpremixed chaotic fire of the hydrogen-air downward injector system using the generalized finite rate chemistry model.…”

Section: Introductionmentioning

confidence: 99%

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Natural Transform along with HPM Technique for Solving Fractional ADE

Pareek¹,

Gupta²,

Agarwal

et al. 2021

Advances in Mathematical Physics

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The authors of this paper solve the fractional space-time advection-dispersion equation (ADE). In the advection-dispersion process, the solute movement being nonlocal in nature and the velocity of fluid flow being nonuniform, it leads to form a heterogeneous system which approaches to model the same by means of a fractional ADE which generalizes the classical ADE, where the time derivative is substituted through the Caputo fractional derivative. For the study of such fractional models, various numerical techniques are used by the researchers but the nonlocality of the fractional derivative causes high computational expenses and complex calculations so the challenge is to use an efficient method which involves less computation and high accuracy in solving such models numerically. Here, in order to get the FADE solved in the form of convergent infinite series, a novel method NHPM (natural homotopy perturbation method) is applied which couples Natural transform along with the homotopy perturbation method. The homotopy peturbation method has been applied in mathematical physics to solve many initial value problems expressed in the form of PDEs. Also, the HPM has an advantage over the other methods that it does not require any discretization of the domains, is independent of any physical parameters, and only uses an embedding parameter p ∈ 0 , 1 . The HPM combined with the Natural transform leads to rapidly convergent series solutions with less computation. The efficacy of the used method is shown by working out some examples for time-fractional ADE with various initial conditions using the NHPM. The Mittag-Leffler function is used to solve the fractional space-time advection-dispersion problem, and the impact of changing the fractional parameter α on the solute concentration is shown for all the cases.

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“…where b:c denotes the ceiling function. This is the same as obtained by Hikal and Abu Ibrahim [12] using ADM.…”

Section: Advances In Mathematical Physicssupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Natural Transform along with HPM Technique for Solving Fractional ADE

Pareek¹,

Gupta²,

Agarwal

et al. 2021

Advances in Mathematical Physics

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“…The solution of FRADE via ADM, in both space and time (STFRADE) with constant diffusion and advection terms, has already been obtained in [62]. As stated, exclusively a temporal dependence for these functions has been considered here in order to model − u and − K in (21), being that the heterogeneity of the medium has already been caught via the fractional approach.…”

Section: The Fractional Advection-diffusion Equation Model and Its So...mentioning

confidence: 99%

“…In this paragraph, the formulation of [62] has been chosen as reference for calibration, where an ADM solution for fractional Advection-Diffusion Equations with constant advective and diffusive coefficients has been found.…”

Section: Calibrationmentioning

confidence: 99%

Modelling Fractional Advection–Diffusion Processes via the Adomian Decomposition

ANTONINI

Salomoni

2023

Mathematics

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When treating geomaterials, fractional derivatives are used to model anomalous dispersion or diffusion phenomena that occur when the mass transport media are anisotropic, which is generally the case. Taking into account anomalous diffusion processes, a revised Fick’s diffusion law is to be considered, where the fractional derivative order physically reflects the heterogeneity of the soil medium in which the diffusion phenomena take place. The solutions of fractional partial differential equations can be computed by using the so-called semi-analytical methods that do not require any discretization and linearization in order to obtain accurate results, e.g., the Adomian Decomposition Method (ADM). Such a method is innovatively applied for overcoming the critical issue of geometric nonlinearities in coupled saturated porous media and the potentialities of the approach are studied, as well as findings discussed.

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“…Momani and Odibat [23] used the ADM and variational iteration approach to solve the space-time fractional ADE. In this aspect, Yildirim and Kocak [29] use the homotopy perturbation methodology in Caputo sense to solve the space-time fractional ADE, whereas Hikal and Abu Ibrahim [30] use the Adomian decomposition method. Using the generalized finite rate chemistry model, Alliche and Chikh [31] investigated the nonpremixed chaotic fire of the hydrogen-air downward injector system.…”

Section: Introductionmentioning

confidence: 99%

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

Aljahdaly

Shah

Naeem

et al. 2022

Journal of Function Spaces

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The approximate solutions of the time fractional advection-dispersion equation are presented in this article. The nonlocal nature of solute movement and the nonuniformity of fluid flow velocity in the advection-dispersion process lead to the formation of a heterogeneous system, which can be modeled using a fractional advection-dispersion equation, which generalizes the classical advection-dispersion equation and replaces the time derivative with the fractional Caputo derivative. Researchers use a variety of numerical techniques to study such fractional models, but the nonlocality of the derivative having fractional order leads to high computation complexity and complex calculations, so the task is to find an efficient technique that requires less computation and provides greater accuracy when numerically solving such models. A innovative techniques, homotopy perturbation method and new iteration method, are used in connection with the Elzaki transform to solve the “fractional advection-dispersion equation” which provides the solution in the convergent series form. When the homotopy perturbation method is used with the Elzaki transform, fast convergent series solutions can be obtained with less computation. By solving some cases of time-fractional advection-dispersion equation with varied initial conditions with the help of new iterative transform method and homotopy perturbation transform method demonstrates the usefulness of the proposed methods.

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On Adomian's Decomposition Method for Solving a Fractional Advection-Dispersion Equation

Cited by 3 publications

References 24 publications

Natural Transform along with HPM Technique for Solving Fractional ADE

Natural Transform along with HPM Technique for Solving Fractional ADE

Modelling Fractional Advection–Diffusion Processes via the Adomian Decomposition

A Comparative Analysis of Fractional Space-Time Advection-Dispersion Equation via Semi-Analytical Methods

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