Laplace equations on the fractal cubes and Casimir effect

Golmankhaneh, Alireza Khalili; Nia, S. Measoomy

doi:10.1140/epjs/s11734-021-00317-4

Cited by 14 publications

(3 citation statements)

References 42 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Additionally, Laplacian and vector differential operators linked with F α -derivatives on fractals were established [50]. Fractal calculus has been extended to include Cantor cubes and Cantor tartan [51], and the Laplace equation has been defined within this framework [52]. The paper is structured as follows: In Section 2, a comparative and review analysis of fractal calculus is presented, focusing on its application to both fractal sets and curves.…”

Section: Introductionmentioning

confidence: 99%

Non-standard analysis for fractal calculus

Golmankhaneh

Welch

Serpa

et al. 2023

J Anal

Self Cite

View full text Add to dashboard Cite

In this paper, we delve into the fascinating realm of fractal calculus applied to fractal sets and fractal curves. Our study includes an exploration of the method analogues of the separable method and the integrating factor technique for solving α-order differential equations. Notably, we extend our analysis to solve Fractal Bernoulli differential equations. The applications of our findings are then showcased through the solutions of problems such as fractal compound interest, the escape velocity of the earth in fractal space and time, and the estimation of time of death incorporating fractal time. Visual representations of our results are also provided to enhance understanding.

show abstract

Section: Introductionmentioning

confidence: 99%

Non-standard analysis for fractal calculus

Golmankhaneh

Welch

Serpa

et al. 2023

J Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…However, validation of non-Darcian models seems to be an endless challenge because alternative models lead to very different solutions of the same problems for fractal porous media. That is why the generalizations from conventional to fractional calculus use standard measures [21] such as length which is not a suitable measure for fractal sets [29]. A more complete model of physical phenomena must include scaling laws and fractality in fractional mechanics [30,31].…”

Section: Introductionmentioning

confidence: 99%

A Mechanical Picture of Fractal Darcy’s Law

Damián Adame,

Gutiérrez-Torres,

Figueroa-Espinoza

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

The main goal of this manuscript is to generalize Darcy’s law from conventional calculus to fractal calculus in order to quantify the fluid flow in subterranean heterogeneous reservoirs. For this purpose, the inherent features of fractal sets are scrutinized. A set of fractal dimensions is incorporated to describe the geometry, morphology, and fractal topology of the domain under study. These characteristics are known through their Hausdorff, chemical, shortest path, and elastic backbone dimensions. Afterward, fractal continuum Darcy’s law is suggested based on the mapping of the fractal reservoir domain given in Cartesian coordinates xi into the corresponding fractal continuum domain expressed in fractal coordinates ξi by applying the relationship ξi=ϵ0(xi/ϵ0)αi−1, which possesses local fractional differential operators used in the fractal continuum calculus framework. This generalized version of Darcy’s law describes the relationship between the hydraulic gradient and flow velocity in fractal porous media at any scale including their geometry and fractal topology using the αi-parameter as the Hausdorff dimension in the fractal directions ξi, so the model captures the fractal heterogeneity and anisotropy. The equation can easily collapse to the classical Darcy’s law once we select the value of 1 for the alpha parameter. Several flow velocities are plotted to show the nonlinearity of the flow when the generalized Darcy’s law is used. These results are compared with the experimental data documented in the literature that show a good agreement in both high-velocity and low-velocity fractal Darcian flow with values of alpha equal to 0<α1<1 and 1<α1<2, respectively, whereas α1=1 represents the standard Darcy’s law. In that way, the alpha parameter describes the expected flow behavior which depends on two fractal dimensions: the Hausdorff dimension of a porous matrix and the fractal dimension of a cross-section area given by the intersection between the fractal matrix and a two-dimensional Cartesian plane. Also, some physical implications are discussed.

show abstract

“…By taking time as a fractal set, the radioactive decay is mathematically modeled and it is shown that the model is Hyers-Ulam stable. Further, fractal calculus is applied to fractal Cantor cubes [ 29 ]. Fractal calculus is defined on the fractal cubes by making use of the integral staircase function.…”

mentioning

confidence: 99%