Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

Liu, Yanqin

doi:10.1155/2012/752869

Cited by 22 publications

(22 citation statements)

References 31 publications

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“…We will outline the main steps of the modified homotopy perturbation method [13][14][15]. For a given time-fractional non-linear non-homogeneous partial differential equation of the form:…”

Section: Caputo Derivative and The Modified Homotopy Perturbation Methodsmentioning

confidence: 99%

“…But only in recent years have fractional differential equations been gained much attention due to their exact description and extensive applications in various of scientific fields from physics to biology, chemistry, and engineering etc. As Liu [13] pointed out that these methods have their deficiencies like the calculation of Adomian polynomials, the Lagrange multiplier, and huge computation work. But we always encounter some difficulty in finding the exact analytical solutions of these problems.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to use the modified homotopy perturbation method [13][14][15], which unifies homotopy perturbation method and Laplace transform method, to solve the heat transfer and porous media equations. The method provides the solutions in a rapid convergent series which may lead to the solution in a closed form.…”

Section: Introductionmentioning

confidence: 99%

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Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations

Yang

2013

THERM SCI

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“…We will outline the main steps of the modified homotopy perturbation method [13][14][15]. For a given time-fractional non-linear non-homogeneous partial differential equation of the form:…”

Section: Caputo Derivative and The Modified Homotopy Perturbation Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations

Yang

2013

THERM SCI

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“…In order to deal with fractional derivatives, we rely on the Caputo operator and on the Grünwald-Letnikov method to numerically approximate the fractional derivatives [28]. Even though various numerical approaches to different types of fractional diffusion models are increasingly appearing in the literature [28][29][30][31][32][33][34], we select this numerical method because it is computationally less expensive than others, and our main aim here is to test the fractional model and not to study the numerical methods for fractional differential equation systems [23,28].…”

Section: Introductionmentioning

confidence: 99%

A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1)

González-Parra

Arenas

Chen-Charpentier

2013

Math. Meth. Appl. Sci.

146

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In this paper, we propose a nonlinear fractional order model in order to explain and understand the outbreaks of influenza A(H1N1). In the fractional model, the next state depends not only upon its current state but also upon all of its historical states. Thus, the fractional model is more general than the classical epidemic models. In order to deal with the fractional derivatives of the model, we rely on the Caputo operator and on the Grünwald–Letnikov method to numerically approximate the fractional derivatives. We conclude that the nonlinear fractional order epidemic model is well suited to provide numerical results that agree very well with real data of influenza A(H1N1) at the level population. In addition, the proposed model can provide useful information for the understanding, prediction, and control of the transmission of different epidemics worldwide. Copyright © 2013 John Wiley & Sons, Ltd.

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“…In 2007, Momani and Odibat (2007) adopted the method for fractional differential equations with great success, and now it is an effective method for fractional calculus (Ganji et al, 2008;Gupta et al, 2012;Jafari and Momani, 2007;Liu, 2012;Madani et al, 2012;Odibat and Momani, 2008). In 2010, the Laplace transform and He's polynomials were used in the homotopy perturbation method (Khan and Mohyud-Din, 2010).…”

Section: Homotopy Perturbation Methodsmentioning

confidence: 99%

Fractional calculus for nanoscale flow and heat transfer

Liu

2014

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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If you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service information about how to choose which publication to write for and submission guidelines are available for all. Please visit www.emeraldinsight.com/authors for more information. About Emerald www.emeraldinsight.comEmerald is a global publisher linking research and practice to the benefit of society. The company manages a portfolio of more than 290 journals and over 2,350 books and book series volumes, as well as providing an extensive range of online products and additional customer resources and services.Emerald is both COUNTER 4 and TRANSFER compliant. The organization is a partner of the Committee on Publication Ethics (COPE) and also works with Portico and the LOCKSS initiative for digital archive preservation. AbstractPurpose -Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach -This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss' divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given. Findings -Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations. Originality/value -Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

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Approximate Solutions of Fractional Nonlinear Equations Using Homotopy Perturbation Transformation Method

Cited by 22 publications

References 31 publications

Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations

Modified homotopy perturbation method coupled with Laplace transform for fractional heat transfer and porous media equations

A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1)

Fractional calculus for nanoscale flow and heat transfer

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