Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

Elbeleze, Asma Ali; Kılıçman, Adem; Taib, Bachok M.

doi:10.1155/2014/803902

Cited by 14 publications

(6 citation statements)

References 16 publications

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“…Other works that prove that convergence of the method are the studies of Biazar and Aminikhah [32] and Elbeleze et al. [33]. HPM presented for a non‐linear and coupled differential equation is presented as follows:…”

Section: Methods Of Solutionmentioning

confidence: 99%

“…The method was first used by He [28], He [29], He [30], and He [31] to solve linear, nonlinear, and coupled boundary value problems in partial or ordinary form. Other works that prove that convergence of the method are the studies of Biazar and Aminikhah [32] and Elbeleze et al [33]. HPM presented for a non-linear and coupled differential equation is presented as follows: Consider a general differential equation defined on a domain…”

Section: Methods Of Solutionmentioning

confidence: 99%

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Effects of viscous and darcy dissipation on fully developed natural convection flow in a composite channel partially filled with porous material: Homotopy perturbation method (HPM)

2023

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A steady natural convection flow in a composite channel partially filled with porous material is investigated analytically in this paper. In this respect, the effects of heat generation due to viscous dissipation and darcy dissipation were considered in the energy equations of the porous region and that of clear fluid region. In addition, the Brinkmann extended Darcy model was applied to model the flow in the porous region. The coupled system of nonlinear ordinary differential equations that governed the flow is solved by the homotopy perturbation method. The analytical solution results for velocity, temperature, and Nusselt number are provided and analyzed. The investigation reveals that increase in the thickness of the porous region hampers the hydrodynamics while it promotes the thermodynamics within the composite channel.

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Section: Methods Of Solutionmentioning

confidence: 99%

Section: Methods Of Solutionmentioning

confidence: 99%

Effects of viscous and darcy dissipation on fully developed natural convection flow in a composite channel partially filled with porous material: Homotopy perturbation method (HPM)

2023

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“…The series in equation ( 7) is convergent in most cases, and the convergence rate of the series depends on the nonlinear operator S. The study of convergence order for a finite series approximation was investigated in Elbeleze et al (2014).…”

Section: Basic Idea Of Homotopy Perturbation Methodsmentioning

confidence: 99%

The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

Nadeem

2021

HFF

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Purpose The purpose of this paper is to find an approximate solution of a fractional differential equation. The fractional Newell–Whitehead–Segel equation (FNWSE) is used to elucidate the solution process, which is one of the nonlinear amplitude equation, and it enhances a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion and convection systems. Design/methodology/approach In Part 1, the authors adopted Mohand transform to find the analytical solution of FNWSE. In this part, the authors apply the fractional complex transform (the two-scale transform) to convert the problem into its differential partner, and then they introduce the homotopy perturbation method (HPM) to bring down the nonlinear terms for the approximate solution. Findings The HPM makes numerical simulation for the fractional differential equations easy, and the two-scale transform is a strong tool for fractal models. Originality/value The HPM with the two-scale transform sheds a bright light on numerical approach to fractional calculus.

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“…Actually, the series is convergent for most cases [17]. Moreover, there are many references discussing the convergence of series approximate solutions by the homotopy perturbation method, one can see [31] for example.…”

Section: Now We Construct the Homotopy Perturbation Equation As Followsmentioning

confidence: 99%

Series Solutions of High-Dimensional Fractional Differential Equations

2021

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In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α.

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Note on the Convergence Analysis of Homotopy Perturbation Method for Fractional Partial Differential Equations

Cited by 14 publications

References 16 publications

Effects of viscous and darcy dissipation on fully developed natural convection flow in a composite channel partially filled with porous material: Homotopy perturbation method (HPM)

Effects of viscous and darcy dissipation on fully developed natural convection flow in a composite channel partially filled with porous material: Homotopy perturbation method (HPM)

The homotopy perturbation method for fractional differential equations: part 2, two-scale transform

Series Solutions of High-Dimensional Fractional Differential Equations

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