A continuum framework for mechanics of fractal materials I: from fractional space to continuum with fractal metric

2015 ◽  
Vol 88 (4) ◽  
Author(s):  
Alexander S. Balankin
Keyword(s):  
Fractal Materials ◽  
Fractional Space

scholarly journals Fractional governing equations of transient groundwater flow in unconfined aquifers with multi-fractional dimensions in fractional time

2020 ◽  
Vol 11 (1) ◽  
pp. 1-12 ◽  
Author(s):  
M. Levent Kavvas ◽  
Tongbi Tu ◽  
Ali Ercan ◽  
James Polsinelli
Keyword(s):  
Groundwater Flow ◽  
Governing Equation ◽  
Water Flux ◽  
Unconfined Aquifer ◽  
Saturated Porous Media ◽  
Unconfined Aquifers ◽  
Heavy Tailed ◽  
Consistent Equation ◽  
Fractional Dimensions ◽  
Fractional Space

Abstract. In this study, a dimensionally consistent governing equation of transient unconfined groundwater flow in fractional time and multi-fractional space is developed. First, a fractional continuity equation for transient unconfined groundwater flow is developed in fractional time and space. For the equation of groundwater motion within a multi-fractional multidimensional unconfined aquifer, a previously developed dimensionally consistent equation for water flux in unsaturated/saturated porous media is combined with the Dupuit approximation to obtain an equation for groundwater motion in multi-fractional space in unconfined aquifers. Combining the fractional continuity and groundwater motion equations, the fractional governing equation of transient unconfined aquifer flow is then obtained. Finally, two numerical applications to unconfined aquifer groundwater flow are presented to show the skills of the proposed fractional governing equation. As shown in one of the numerical applications, the newly developed governing equation can produce heavy-tailed recession behavior in unconfined aquifer discharges.

Gravitational potential in fractional space

Open Physics ◽  
2007 ◽  
Vol 5 (3) ◽  
Author(s):  
Sami Muslih ◽  
Dumitru Baleanu ◽  
Eqab Rabei
Keyword(s):  
Mass Distribution ◽  
Gravitational Potential ◽  
Universal Constant ◽  
Fractional Mass ◽  
Fractional Space

AbstractIn this paper the gravitational potential with β-th order fractional mass distribution was obtained in α dimensionally fractional space. We show that the fractional gravitational universal constant G α is given by $$G_\alpha = \frac{{2\Gamma \left( {\frac{\alpha }{2}} \right)}}{{\pi ^{\alpha /2 - 1} (\alpha - 2)}}G$$ , where G is the usual gravitational universal constant and the dimensionality of the space is α > 2.

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