Nonlocal Models for Transport in Fractal Media

Fractals ◽  
2017 ◽  
pp. 153-168
Author(s):  
Daniel O’Malley ◽  
John H. Cushman
Keyword(s):  
Fractal Media ◽  
Nonlocal Models

An Overview: Analysis of Heat and Mass Transfer in Fractal Media by Fractal Geometry and Technique

2010 ◽  
Author(s):  
Boming Yu
Keyword(s):  
Heat Transfer ◽  
Mass Transfer ◽  
Porous Media ◽  
Heat And Mass Transfer ◽  
Oil Recovery ◽  
Fractal Geometry ◽  
Boiling Heat Transfer ◽  
Nucleate Boiling ◽  
Fractal Media ◽  
Nucleate Boiling Heat Transfer

In the past three decades, fractal geometry and technique have received considerable attention due to its wide applications in sciences and technologies such as in physics, mathematics, geophysics, oil recovery, material science and engineering, flow and heat and mass transfer in porous media etc. The fractal geometry and technique may become particularly powerful when they are applied to deal with random and disordered media such as porous media, nanofluids, nucleate boiling heat transfer. In this paper, a summary of recent advances is presented in the areas of heat and mass transfer in fractal media by fractal geometry technique. The present overview includes a brief summary of the fractal geometry technique applied in the areas of heat and mass transfer; thermal conductivities of porous media and nanofluids; nucleate boiling heat transfer. A few comments are made with respect to the theoretical studies that should be made in the future.

scholarly journals Magnetohydrodynamics of fractal media

2006 ◽  
Vol 13 (5) ◽  
pp. 052107 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractal Media

Continuum Homogenization of Fractal Media

2019 ◽  
pp. 905-935
Author(s):  
Martin Ostoja-Starzewski ◽  
Jun Li ◽  
Paul N. Demmie
Keyword(s):  
Fractal Media

scholarly journals A PRIORI ESTIMATE FOR AN EQUATION WITH FRACTIONAL DERIVATIVES WITH DIFFERENT ORIGINS

2019 ◽  
pp. 41-47
Author(s):  
Л.М. Энеева
Keyword(s):  
Fractional Derivatives ◽  
Model Equation ◽  
A Priori ◽  
A Priori Estimate ◽  
Equation Of Motion ◽  
Point Boundary ◽  
Priori Estimate ◽  
Fractal Media ◽  
Different Origins ◽  
Variable Potential

В работе исследуется обыкновенное дифференциальное уравнение дробного порядка, содержащее композицию дробных производных с различными началами, с переменным потенциалом. Рассматриваемое уравнение выступает модельным уравнением движения во фрактальной среде. Для исследуемого уравнения доказана априорная оценка решения смешанной двухточечной краевой задачи. We consider an ordinary differential equation of fractional order with the composition of leftand right-sided fractional derivatives, and with variable potential. The considered equation is a model equation of motion in fractal media. We prove an a priori estimate for solutions of a mixed two-point boundary value problem for the equation under study.

scholarly journals Analytical solution to local fractional Landau-Ginzburg-Higgs equation on fractal media

2021 ◽  
Vol 25 (6 Part B) ◽  
pp. 4449-4455
Author(s):  
Shu-Xian Deng ◽  
Xin-Xin Ge
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Present Article ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Variational Iteration ◽  
Fractal Media

The main objective of the present article is to introduce a new analytical solution of the local fractional Landau-Ginzburg-Higgs equation on fractal media by means of the local fractional variational iteration transform method, which is coupling of the variational iteration method and Yang-Laplace transform method.

scholarly journals A Derivation of Proca Equations on Cantor Sets: A Local Fractional Approach

2014 ◽  
Vol 10 ◽  
pp. 48-56 ◽  
Author(s):  
Victor Christianto ◽  
Biruduganti Rahul
Keyword(s):  
Maxwell Equations ◽  
High Energy Physics ◽  
Stokes Equations ◽  
High Energy ◽  
Cantor Sets ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fractal Media ◽  
Vector Calculus ◽  
Proca Equations

In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Using the same approach, elsewhere Yang, Baleanu & Tenreiro Machado derived systems of Navier-Stokes equations on Cantor sets. However, so far there is no derivation of Proca equations on Cantor sets. Therefore, in this paper we present for the first time a derivation of Proca equations and GravitoElectroMagnetic (GEM) Proca-type equations on Cantor sets. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, We suggest that Proca equations on Cantor sets can describe electromagnetic of fractal superconductors; besides GEM Proca-type equations on Cantor sets may be used to explain some gravitoelectromagnetic effects of superconductor for fractal media. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions

2018 ◽  
Vol 39 (2) ◽  
pp. 607-625 ◽  
Author(s):  
Qiang Du ◽  
Yunzhe Tao ◽  
Xiaochuan Tian ◽  
Jiang Yang
Keyword(s):  
Green's Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Optimal Order ◽  
Numerical Approximations ◽  
Splitting Technique ◽  
Singular Measures ◽  
Nonlocal Models

AbstractNonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green’s functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green’s functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures.

Carbon Nanotube Interconnects with Air-Gaps: Effect on Thermal Stability, Delay and Area

2017 ◽  
Vol 48 ◽  
pp. 29-37 ◽  
Author(s):  
P. Uma Sathyakam ◽  
Partha S. Mallick
Keyword(s):  
Carbon Nanotube ◽  
Differential Equations ◽  
Dielectric Medium ◽  
Single Walled Carbon Nanotube ◽  
Chip Area ◽  
Air Gaps ◽  
Ambient Temperatures ◽  
Performance Factors ◽  
Fractal Media ◽  
Technology Nodes

This paper presents single walled carbon nanotube (SWCNT) interconnects with air as dielectric medium. We treat CNT interconnects as a discrete (fractal) media for the first time where continuum based differential equations fail to capture the physics at nanoscale and hence, we use discrete partial differential equations in this work. We have analyzed the effect of air gaps (AG) on performance factors like temperature dependent resistance R(T) of CNTs and hence the R(T)C delay of the interconnects. We have first calculated the temperature coefficient of resistance (TCR) of CNTs and analyzed the trend of changing resistance at different ambient temperatures. The R(T)C delay shows that CNT/AG interconnects can operate satisfactorily up to 500K. We then compare the R(T)C delay with ITRS predictions from 17nm to 8nm technology nodes. We have also calculated the chip area used by CNT/air-gap interconnects and found that they take up to 83% lesser area than the conventional Cu/low-k interconnects.

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