A fractal derivative model for snow’s thermal insulation property

Wang, Yan; Yao, Shao-Wen; Yang, Hongwei

doi:10.2298/tsci1904351w

Cited by 55 publications

(34 citation statements)

References 11 publications

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“…2Using actual data of (1.1)-(1.7), the authors try to accomplish numerical tests and give a professional software for environmental treatment. 3Studying the recent research on variational formulation and fractal calculus, this work may possibly be continued for the contamination transport [44][45][46][47][48][49][50][51][52].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

A Mixed Volume Element With Characteristic Mixed Volume Element for Contamination Transport Problem

Yuan¹,

Li²,

Song³

et al. 2020

Jour. Pure Appl. Math. Adv. Appl.

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Nonlinear systems of convection-dominated diffusion equations are used as the mathematical model of contamination transport problem which is an important topic in environmental protection science. An elliptic equation defines the pressure, a convection-diffusion equation expresses the concentration of contamination, and an ordinary differential equation interprets the surface absorption concentration. The transport pressure appears in the equation of the concentration which determines the Darcy velocity and also controls the physical process. The method of conservative mixed volume element is used to solve the flow equation which improves the computational accuracy of Darcy velocity by one order. We use the mixed volume element with the characteristic to approximate the concentration. This method of characteristic not only preserves the strong computational stability at sharp front, but also eliminates numerical dispersion and nonphysical oscillation. In the present scheme, we could adopt a large step without losing accuracy. The diffusion is approximated by the mixed volume element. The concentration and its adjoint vector function are obtained simultaneously, and the locally conservative law is preserved. We derive an optimal second order estimates in norm-2 l by using the theory and technique of a priori estimates of a differential equation. From the numerical examples given in the paper, the method shows its potential to be a powerful tool in solving actual problems.

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Section: Conclusion and Discussionmentioning

confidence: 99%

A Mixed Volume Element With Characteristic Mixed Volume Element for Contamination Transport Problem

Yuan¹,

Li²,

Song³

et al. 2020

Jour. Pure Appl. Math. Adv. Appl.

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“…This transform makes the fractal calculus extremely simple in view of traditional calculus. Now the fractal calculus has been applied to non-linear vibration [62], biomechanics [63][64][65], electrochemical arsenic sensor [66], tsunami model [67], thermal insulation [68], fractal rate model [69], biomimic design [70,71], fractal diffusion [72], fractal filtration [73], and nanotechnology [74][75][76][77][78].…”

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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“…This paradox can be solved using the twoscale thermodynamics. 20,21 Actually the inner surface is not smooth enough (see Figure 1), so a continuum model with smooth boundary on a large scale leads to a wrong result; however, if we observe the problem on a smaller scale so that the unsmooth inner surface can be measured, the paradox can be completely solved by the fractal calculus, 22,23 which is to study various phenomena in discontinuous space, and has widely applied in electrochemistry, 24 biomechanics, 25,26 Tsunami model, 27 wool fiber, 28 thermal insulation, 29 fractal solitary wave, 30 and fractal convection-diffusion model. 31 As shown in Figure 1, the inner surface of the hollow fiber is not smooth, and the inner boundary of the cross section is a coastline-like curve with fractal dimensions larger than 1, so the fractal calculus has to be adopted in our study.…”

Section: Fractal Diffusionmentioning

confidence: 99%

Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface

Yao

2019

Journal of Engineered Fibers and Fabrics

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A fractal derivative model for snow’s thermal insulation property

Cited by 55 publications

References 11 publications

A Mixed Volume Element With Characteristic Mixed Volume Element for Contamination Transport Problem

A Mixed Volume Element With Characteristic Mixed Volume Element for Contamination Transport Problem

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

Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface

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