The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors

Abstract

a b s t r a c tIn a non-local fractional-order model of thermal energy transport recently introduced by the authors, it is assumed that local and non-local contributions coexist at a given observation scale: while the first is described by the classical Fourier transport law, the second involves couples of adjacent and non-adjacent elementary volumes, and is taken as proportional to the product of the masses of the interacting volumes and their relative temperature, through a material-dependent, distance-decaying power-law function. As a result, a fractional-order heat conduction equation is derived. This paper presents a pertinent finite element method for the solution of the proposed fractional-order heat conduction equation. Homogenous and non-homogeneous rigid bodies are considered. Numerical applications are carried out on 1D and 2D bodies, including a standard finite difference solution for validation.

Advanced materials modelling via fractional calculus: challenges and perspectives

“…Heat conduction is a typical phenomenon where time non-locality of fractional operators has been used to model memory effects. Various time-fractional heat conduction equations have been proposed, differing each other for the fractional operators involved [56][57][58][59][60][61]. In this framework, a thermodynamic model corresponding to power-law rise of temperature and heat flux has been proposed recently by Zingales [62].…”

Using Fractal Calculus to Solve Fractal Navier–Stokes Equations, and Simulation of Laminar Static Mixing in COMSOL Multiphysics

“…The fractional calculus came to their aid as the non-local theory. Several fractional models have been introduced by researchers following this approach, for instance [3][4][5][6]. In other words, many believe that substituting a non-local theory instead of a local theory is the only solution.…”

Existence and Stability Analysis for Fractional Impulsive Caputo Difference-Sum Equations with Periodic Boundary Condition

“…. , T}; t k ∈ N α,T+α−1 and Fractional calculus has been gaining more attention over the past decade as this calculus has been addressed to various problems used in science and engineering; see [1][2][3][4][5][6][7][8]. For fractional difference calculus theory, which is the discrete case of fractional calculus, there is still not much interest among mathematical researchers.…”