Fractional Calculus Applied to Fractal Media and Nonlocal Continua

2021 ◽  
pp. 689-728
Author(s):  
Alberto Carpinteri
Keyword(s):  
Fractional Calculus ◽  
Fractal Media

scholarly journals Electro-chemical manifestation of nanoplasmonics in fractal media

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Emmanuel Baskin ◽  
Alexander Iomin
Keyword(s):  
Experimental Data ◽  
Composite Materials ◽  
Fractional Calculus ◽  
Fractal Geometry ◽  
Local Enhancement ◽  
Plasmon Excitation ◽  
Fractal Media ◽  
Surface Plasmon Excitation ◽  
The Media ◽  
Fractional Integrodifferentiation

AbstractElectrodynamics of composite materials with fractal geometry is studied in the framework of fractional calculus. This consideration establishes a link between fractal geometry of the media and fractional integrodifferentiation. The photoconductivity in the vicinity of the electrode-electrolyte fractal interface is studied. The methods of fractional calculus are employed to obtain an analytical expression for the giant local enhancement of the optical electric field inside the fractal composite structure at the condition of the surface plasmon excitation. This approach makes it possible to explain experimental data on photoconductivity in the nano-electrochemistry.

scholarly journals Local Fractional Poisson and Laplace Equations with Applications to Electrostatics in Fractal Domain

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yang-Yang Li ◽  
Yang Zhao ◽  
Gong-Nan Xie ◽  
Dumitru Baleanu ◽  
Xiao-Jun Yang ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Cylindrical Coordinates ◽  
Fractal Media ◽  
Laplace Equations ◽  
Local Fractional Calculus ◽  
Fractal Domain

From the local fractional calculus viewpoint, Poisson and Laplace equations were presented in this paper. Their applications to the electrostatics in fractal media are discussed and their local forms in the Cantor-type cylindrical coordinates are also obtained.

scholarly journals On the fractal heat transfer problems with local fractional calculus

2015 ◽  
Vol 19 (5) ◽  
pp. 1867-1871 ◽  
Author(s):  
Duan Zhao ◽  
Xiao-Jun Yang ◽  
H.M. Srivastava
Keyword(s):  
Heat Transfer ◽  
Fractional Calculus ◽  
Nonlinear Oscillator ◽  
Fractal Media ◽  
Local Fractional Calculus ◽  
Transfer Equations ◽  
Linear And Nonlinear ◽  
Transfer Problems ◽  
Homogeneous Linear ◽  
Oscillator Equations

This article investigates several fractal heat transfer problems from the local fractional calculus viewpoint. At low and high excess temperatures, the linear and nonlinear heat-transfer equations are presented. The non-homogeneous linear and nonlinear oscillator equations in fractal heat transfer are discussed. The results are adopted to present the behaviors of the heat transfer in fractal media.

Fractional calculus for nanoscale flow and heat transfer

2014 ◽  
Vol 24 (6) ◽  
pp. 1227-1250 ◽  
Author(s):  
Hong-Yan Liu ◽  
Ji-Huan He ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Content Type ◽  
Variational Iteration ◽  
Fractal Media ◽  
Fractional Differential ◽  
Analytical Approaches

Purpose – Academic and industrial researches on nanoscale flows and heat transfers are an area of increasing global interest, where fascinating phenomena are always observed, e.g. admirable water or air permeation and remarkable thermal conductivity. The purpose of this paper is to reveal the phenomena by the fractional calculus. Design/methodology/approach – This paper begins with the continuum assumption in conventional theories, and then the fractional Gauss’ divergence theorems are used to derive fractional differential equations in fractal media. Fractional derivatives are introduced heuristically by the variational iteration method, and fractal derivatives are explained geometrically. Some effective analytical approaches to fractional differential equations, e.g. the variational iteration method, the homotopy perturbation method and the fractional complex transform, are outlined and the main solution processes are given. Findings – Heat conduction in silk cocoon and ground water flow are modeled by the local fractional calculus, the solutions can explain well experimental observations. Originality/value – Particular attention is paid throughout the paper to giving an intuitive grasp for fractional calculus. Most cited references are within last five years, catching the most frontier of the research. Some ideas on this review paper are first appeared.

scholarly journals REVIEW OF SOME PROMISING FRACTIONAL PHYSICAL MODELS

2013 ◽  
Vol 27 (09) ◽  
pp. 1330005 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fractional Calculus ◽  
Power Law ◽  
Particle Interaction ◽  
Discrete Systems ◽  
Physical Models ◽  
Long Term Memory ◽  
Fractional Dynamics ◽  
Fractal Media ◽  
Systems With Memory ◽  
With Memory

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of non-integer orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for future development of fractional dynamics. We suggest a short introduction to fractional calculus as a theory of integration and differentiation of noninteger order. Some applications of integro-differentiations of fractional orders in physics are discussed. Models of discrete systems with memory, lattice with long-range inter-particle interaction, dynamics of fractal media are presented. Quantum analogs of fractional derivatives and model of open nano-system systems with memory are also discussed.

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