He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy

Liu, Fujuan; Li, Zheng-Biao; Zhang, Sheng; Liu, Hongyan

doi:10.2298/tsci1504155l

Cited by 49 publications

(40 citation statements)

References 6 publications

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“…By using the VIM, He gave the fD and fI which involve the terms determined by the initial or boundary condition. Liu et al [16] discussed the solution of heat conduction in a fractal medium with the aid of He's fD. Kumar et al discussed the solution of partial differential equations involving time-fD by using Laplace transform and perturbation method based on the VIM; see [17,18] and references in them.…”

Section: Remarks On Recent Developmentsmentioning

confidence: 99%

Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform

Morita

Sato

2016

Mathematics

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Abstract:In a series of papers, we discussed the solution of Laplace's differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer's DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann-Liouville fractional derivative (fD) and the distribution theory in the space D R or the AC of Laplace transform. We now obtain the solutions of the hypergeometric DE, which are expressed by the hypergeometric functions, with the aid of the AC of Riemann-Liouville fD, and the distribution theory in the space D r,R , which is introduced in this paper, or by the term-by-term inverse Laplace transform of AC of Laplace transform of the solution expressed by a series.

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Section: Remarks On Recent Developmentsmentioning

confidence: 99%

Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform

Morita

Sato

2016

Mathematics

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“…Very recently, fractional operator, whose derivative has singular kernel introduced by Yang et al [14]. Motivated by above work many researchers applied new derivative in certain real world problems (see, e.g., [5,7,[15][16][17]). In the sequel, we aim to extend the definition of the classical Caputo fractional derivative operator.…”

Section: Extended Caputo Fractional Derivative Operatormentioning

confidence: 99%

An extension of Caputo fractional derivative operator and its applications

Kıymaz¹,

Çetinkaya²,

Agarwa³

2016

J. Nonlinear Sci. Appl.

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In this paper, an extension of Caputo fractional derivative operator is introduced, and the extended fractional derivatives of some elementary functions are calculated. At the same time, extensions of some hypergeometric functions and their integral representations are presented by using the extended fractional derivative operator, linear and bilinear generating relations for extended hypergeometric functions are obtained, and Mellin transforms of some extended fractional derivatives are also determined.

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“…or it can convert a fractional di erential equation to a partial di erential equation as given in [28].…”

Section: Formulation and Analysismentioning

confidence: 99%

Peristaltic transport of a fractional Burgers’ fluid with variable viscosity through an inclined tube

Rachid

2015

Open Physics

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Abstract:In the present study, we investigate the unsteady peristaltic transport of a viscoelastic uid with fractional Burgers' model in an inclined tube. We suppose that the viscosity is variable in the radial direction. This analysis has been carried out under low Reynolds number and long-wavelength approximations. An analytical solution to the problem is obtained using a fractional calculus approach. Figures are plotted to show the e ects of angle of inclination, Reynolds number, Froude number, material constants, fractional parameters, parameter of viscosity and amplitude ratio on the pressure gradient, pressure rise, friction force, axial velocity and on the mechanical efciency.

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He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy

Cited by 49 publications

References 6 publications

Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform

Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform

An extension of Caputo fractional derivative operator and its applications

Peristaltic transport of a fractional Burgers’ fluid with variable viscosity through an inclined tube

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