Search results for "calculus"
showing 10 items of 617 documents
Fractional mechanical model for the dynamics of non-local continuum
2009
In this chapter, fractional calculus has been used to account for long-range interactions between material particles. Cohesive forces have been assumed decaying with inverse power law of the absolute distance that yields, as limiting case, an ordinary, fractional differential equation. It is shown that the proposed mathematical formulation is related to a discrete, point-spring model that includes non-local interactions by non-adjacent particles with linear springs with distance-decaying stiffness. Boundary conditions associated to the model coalesce with the well-known kinematic and static constraints and they do not run into divergent behavior. Dynamic analysis has been conducted and both…
The finite element method for fractional non-local thermal energy transfer in non-homogeneous rigid conductors
2015
Abstract In 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…
A physically based connection between fractional calculus and fractal geometry
2014
We show a relation between fractional calculus and fractals, based only on physical and geometrical considerations. The link has been found in the physical origins of the power-laws, ruling the evolution of many natural phenomena, whose long memory and hereditary properties are mathematically modelled by differential operators of non integer order. Dealing with the relevant example of a viscous fluid seeping through a fractal shaped porous medium, we show that, once a physical phenomenon or process takes place on an underlying fractal geometry, then a power-law naturally comes up in ruling its evolution, whose order is related to the anomalous dimension of such geometry, as well as to the m…
Applications and Implications of Fractional Dynamics for Dielectric Relaxation
2012
This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. …
Fractional-Order Thermal Energy Transport for Small-Scale Engineering Devices
2014
Fractional-order thermodynamics has proved to be an efficient tool to describe several small-scale and/or high-frequency thermodynamic processes, as shown in many engineering and physics applications. The main idea beyond fractional-order physics and engineering relies on replacing the integer-order operators of classical differential calculus with their real-order counterparts. In this study, the authors aim to extend a recently proposed physical picture of fractional-order thermodynamics to a generic 3D rigid heat conductor where the thermal energy transfer is due to two phenomena: a short-range heat flux ruled by stationary and nonstationary transport equations, and a long-range thermal …
Non-Markovian Wave Function Simulations of Quantum Brownian Motion
2005
The non-Markovian wave function method (NMWF) using the stochastic unravelling of the master equation in the doubled Hilbert space is implemented for quantum Brownian motion. A comparison between the simulation and the analytical results shows that the method can be conveniently used to study the non-Markovian dynamics of the system.
Phase space coordinates and the Hamiltonian constraint of Regge calculus.
1994
We suggest that the phase space of Regge calculus is spanned by the areas and the deficit angles corresponding to the two-simplexes on the spacelike hypersurface of simplicial spacetime. Our proposal is based on a slight modification of the Ashtekar formulation of canonical gravity. In terms of these phase space coordinates we write an equation which we suggest to be a simplicial version of the Hamiltonian constraint of canonical gravity.
Construction of the ground state in nonrelativistic QED by continuous flows
2006
AbstractFor a nonrelativistic hydrogen atom minimally coupled to the quantized radiation field we construct the ground state projection Pgs by a continuous approximation scheme as an alternative to the iteration scheme recently used by Fröhlich, Pizzo, and the first author [V. Bach, J. Fröhlich, A. Pizzo, Infrared-finite algorithms in QED: The groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. (2006), doi: 10.1007/s00220-005-1478-3]. That is, we construct Pgs=limt→∞Pt as the limit of a continuously differentiable family (Pt)t⩾0 of ground state projections of infrared regularized Hamiltonians Ht. Using the ODE solved by this family of projections, we sho…