Non-differentiable Solutions for Local Fractional Nonlinear Riccati Differential Equations

Yang, Xiao‐Jun; Srivástava, H. M.; Torres, Delfim F. M.; Zhang, Yudong

doi:10.3233/fi-2017-1500

Cited by 17 publications

(18 citation statements)

References 27 publications

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“…We obtain the family of the nonlinear local fractional ordinary differential equations as follows. Family A When = 1, we have the following nonlinear local fractional ordinary differential equation 23,24 d Ω ( )…”

Section: Some Exact Solutions For the Nonlinear Fractal Burgers' Equamentioning

confidence: 99%

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

Yang

Machado

2019

Math Methods in App Sciences

119

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In this letter, we consider the new nonlinear Burgers' equation engaging local fractional derivative for the first time. With the use of the travelling-wave transformation of non-differentiable type, some exact solutions for the new nonlinear Burgers' equation are discussed in detail. The obtained results are accurate and efficient for the descriptions of the acoustic signals propagation in the fractal stratified media.

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Section: Some Exact Solutions For the Nonlinear Fractal Burgers' Equamentioning

confidence: 99%

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

Yang

Machado

2019

Math Methods in App Sciences

119

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“…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%

“…[1][2][3][4] Many researchers worked to model wave equations from the well-known wave equations by substituting the standard derivatives by the arbitrary order derivative. 5,6 Nowadays, the local fractional calculus 7 is tried to report the nondifferentiable problems, for example, heat conduction problem involving local derivative of fractional order, 7,8 local fractional Tricomi equation, 9 fractal vehicular traffic flow, 10 Laplace equation containing local fractional operator, 11 nonlinear gas dynamics equation, and coupled KdV equation pertaining to local operator of noninteger order, 12 the wave equation involving noninteger order derivative introduced by Yang, 13 the system of partial differential equations with local operator of noninteger order, 14 heat conduction equations with local fractional calculus, 15 nonlinear Riccati differential equations involving local fractional operator, 16 local fractional telegraph equations occurring in electrical transmission line, 17 local fractional LWR equation, 18 local fractional modeling in growths of populations, 19 local fractional model is used in kidney images enhancement, 20 Fitzhugh-Nagumo equations with local fractional derivative, 21 mathematical model of shallow water waves with the aid of local fractional KdV equation, 22 Boussinesq equation containing local fractional operator, 23 local fractional KdV equation, and its exact traveling wave solution, 24 etc.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Fractional calculus has a 300-year-old history, however its applications in physics and engineering were only recently identified. Some very recent papers on its applications are given in [13,16,18,22,26,30,38,39] and the references therein. It was found that many systems in interdisciplinary fields can be elegantly described with the help of fractional derivative such as given in [22] and [30].…”

Section: Introductionmentioning

confidence: 99%

Adaptive add order synchronization and anti-synchronization of fractional order chaotic systems with fully unknown parameters

Al-Sawalha¹,

Ghazel²,

Ababneh³

et al. 2017

J. Nonlinear Sci. Appl.

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In this paper, an adaptive control scheme is developed to study the add order synchronization and the add order antisynchronization behavior between two different dimensional fractional order chaotic systems with fully uncertain parameters. This design of adaptive controller is based on the Lyapunov stability theory. Analytic expression for the controller with its adaptive laws of parameters is shown. The adaptive add order synchronization and add order anti-synchronization between two fractional order chaotic systems are used to show the effectiveness of the proposed method. Theoretical analysis and numerical simulations are used to verify the results.

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Non-differentiable Solutions for Local Fractional Nonlinear Riccati Differential Equations

Cited by 17 publications

References 27 publications

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

A new fractal nonlinear Burgers' equation arising in the acoustic signals propagation

On the local fractional wave equation in fractal strings

Adaptive add order synchronization and anti-synchronization of fractional order chaotic systems with fully unknown parameters

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