A new approximate analytical method for a system of fractional differential equations

Yao, Shao-Wen; Wang, Kang‐Le

doi:10.2298/tsci180613120y

Cited by 6 publications

(10 citation statements)

References 20 publications

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“…On a smaller scale, for example a scale of water molecule's size, water becomes discontinuous and all laws based on continuous space or continuous time become invalid. Generally we can use Mandelbrot's fractal theory [44] to model the discontinuous phenomena [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60]. Newton's calculus is established on an infinitesimal assumption and the function is differentiable, however, the molecule's motion in water at an infinitesimal interval of time or distance is not differentiable.…”

Section: Dimension Is Everything and Two Scale Fractal Geometrymentioning

confidence: 99%

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

2020

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Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed.

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Section: Dimension Is Everything and Two Scale Fractal Geometrymentioning

confidence: 99%

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

2020

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“…Equations (16) and (17) can be written in simpler forms: The variational principle of Equation 11under constraints of Equations (9) and (10) become an optimal problem to minimize J(a,b,c) under the constraints of Equations (18) and (19). The stationary conditions are [27,28] d…”

Section: Fractal Variational Principlementioning

confidence: 99%

Optimization of a fractal electrode-level charge transport model

Liu

2021

THERM SCI

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“…If the one iteration operation is required, one can substitute equations (14) and (19) into the expansion (12) making q ¼ 1 and e ¼ 1; the final solution will offer in the form…”

Section: The Fractional Derivativementioning

confidence: 99%

A periodic solution of the fractional sine-Gordon equation arising in architectural engineering

Shen

El‐Dib

2020

Journal of Low Frequency Noise, Vibration and Active Control

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Many nonlinear vibrations arising in the engineering of architecture include noise and uncertain properties, and this paper suggests a fractional model to elucidate the properties. The fractional sine-Gordon equation with the Riemann-Liouville fractional derivative is used as an example to solve its periodic solution by the homotopy perturbation method. The frequency-amplitude relationship is obtained, and the effect of the fractional derivative order on the vibration property is discussed. Additionally, the harmonic resonance is also discussed. This preliminary research can be further extended to real applications.

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A new approximate analytical method for a system of fractional differential equations

Abstract: In this paper, a new approximate analytical method is established, and it is useful in constructing approximate analytical solution a system of fractional differential equations. The results show that our method is reliable and efficient for solving the fractional system.

Cited by 6 publications

References 20 publications

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

Optimization of a fractal electrode-level charge transport model

A periodic solution of the fractional sine-Gordon equation arising in architectural engineering

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