A new insight into complexity from the local fractional calculus view point: modelling growths of populations

In the present paper, a family of the special functions via the celebrated Mittag-Leffler function defined on the Cantor sets is investigated. The nonlinear local fractional ODEs (NLFODEs) are presented by following the rules of local fractional derivative (LFD). The exact solutions for these problems are also discussed with the aid of the non-differentiable charts on Cantor sets. The obtained results are important for describing the characteristics of the fractal special functions.

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“…The LFD of Π τ (µ) of fractal order τ (0 < τ < 1) at the point µ = µ 0 is defined by 24,[38][39][40][41][44][45][46][47][48] …”

Section: Preliminaries Definitions and Fractal Special Functionsmentioning

confidence: 99%

“…[34][35][36][37] There is an alternative operator (called local FC) to model the local FODEs in fractal electric circuits, 38 free damped vibrations, 39 shallow water surfaces 40 and populations. [41][42][43] The fractal partial differential equations (FPDEs) in mathematical physics were also discussed in Refs. [44][45][46][47][48].…”

Section: Introductionmentioning

confidence: 99%

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

2017

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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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“…Nowadays, the local fractional calculus is tried to report the nondifferentiable problems, for example, heat conduction problem involving local derivative of fractional order, local fractional Tricomi equation, fractal vehicular traffic flow, Laplace equation containing local fractional operator, nonlinear gas dynamics equation, and coupled KdV equation pertaining to local operator of noninteger order, the wave equation involving noninteger order derivative introduced by Yang, the system of partial differential equations with local operator of noninteger order, heat conduction equations with local fractional calculus, nonlinear Riccati differential equations involving local fractional operator, local fractional telegraph equations occurring in electrical transmission line, local fractional LWR equation, local fractional modeling in growths of populations, local fractional model is used in kidney images enhancement, Fitzhugh–Nagumo equations with local fractional derivative, mathematical model of shallow water waves with the aid of local fractional KdV equation, Boussinesq equation containing local fractional operator, local fractional KdV equation, and its exact traveling wave solution, etc.…”

Section: Introductionmentioning

confidence: 99%

“…The local fractional derivative is discussed as

\frac{\partial^{β} v ((),, ξ, η)}{\partial η^{β}} = \frac{Δ^{β} ((),, v ((),, ξ, η) - v true(ξ, η_{0} true))}{(η - η_{0})},

where

Δ^{β} ((),, v ((),, ξ, η) - v true(ξ, η_{0} true)) ≅ normalΓ ((), 1 + β) ([],, v ((),, ξ, η) - v true(ξ, η_{0} true)) .

…”

Section: Introductionmentioning

confidence: 99%

On the local fractional wave equation in fractal strings

Singh

Kumar

Băleanu

et al. 2019

Math Methods in App Sciences

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The key aim of the present study is to attain nondifferentiable solutions of extended wave equation by making use of a local fractional derivative describing fractal strings by applying local fractional homotopy perturbation Laplace transform scheme. The convergence and uniqueness of the obtained solution by using suggested scheme is also examined. To determine the computational efficiency of offered scheme, some numerical examples are discussed. The results extracted with the aid of this technique verify that the suggested algorithm is suitable to execute, and numerical computational work is very interesting.

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A new insight into complexity from the local fractional calculus view point: modelling growths of populations

Cited by 41 publications

References 24 publications

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

Non-Differentiable Exact Solutions for the Nonlinear Odes Defined on Fractal Sets

An extension of Caputo fractional derivative operator and its applications

On the local fractional wave equation in fractal strings

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