Anomalous diffusion based on fractional calculus approach applied to drying analysis of apple slices: The effects of relative humidity and temperature

2017 ◽  
Vol 40 (5) ◽  
pp. e12549 ◽  
Author(s):  
C. Ramírez ◽  
V. Astorga ◽  
H. Nuñez ◽  
A. Jaques ◽  
R. Simpson
Keyword(s):  
Relative Humidity ◽  
Fractional Calculus ◽  
Anomalous Diffusion ◽  
Apple Slices

FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3993-3999
Author(s):  
SUMIYOSHI ABE
Keyword(s):  
Brownian Motion ◽  
Quantum Theory ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Master Equation ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Quantum Master Equation ◽  
Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

scholarly journals Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 790
Author(s):  
Yong Zhang ◽  
Dongbao Zhou ◽  
Wei Wei ◽  
Jonathan M. Frame ◽  
Hongguang Sun ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Broad Spectrum ◽  
Anomalous Diffusion ◽  
Anomalous Transport ◽  
Full Spectrum ◽  
Advection Dispersion ◽  
Unit Scale ◽  
Pollutants Transport ◽  
Spatially Varying ◽  
River Corridor

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology.

scholarly journals Discrete fractional diffusion model with two memory terms

2015 ◽  
Vol 19 (4) ◽  
pp. 1177-1181
Author(s):  
Yan-Mei Qin ◽  
Hua Kong ◽  
Kai-Teng Wu ◽  
Xiao-Ming Zhu
Keyword(s):  
Numerical Simulation ◽  
Fractional Calculus ◽  
Diffusion Model ◽  
Discrete Time ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Difference ◽  
Linear Evolution ◽  
Time Domains ◽  
Diffusion Behaviors

Fractional calculus can always exactly describe anomalous diffusion. Recently the discrete fractional difference is becoming popular due to the depiction of non-linear evolution on discrete time domains. This paper proposes a diffusion model with two terms of discrete fractional order. The numerical simulation is given to reveal various diffusion behaviors.

scholarly journals On Riemann-Liouville and Caputo Derivatives

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Changpin Li ◽  
Deliang Qian ◽  
YangQuan Chen
Keyword(s):  
Signal Processing ◽  
Fractional Calculus ◽  
Anomalous Diffusion ◽  
Caputo Derivative ◽  
Fractional Derivatives ◽  
Science And Engineering ◽  
Caputo Derivatives ◽  
Fractional Equations

Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann-Liouville (RL) derivative, one of mostly used fractional derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in science and engineering.

scholarly journals General fractional calculus in non-singular power-law kernel applied to model anomalous diffusion phenomena in heat transfer problems

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 11-18 ◽  
Author(s):  
Feng Gao
Keyword(s):  
Heat Transfer ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Power Law ◽  
Anomalous Diffusion ◽  
Fourier Transforms ◽  
Diffusion Phenomena ◽  
Complex Phenomena ◽  
Transfer Problems ◽  
First Time

In this paper we address the general fractional calculus of Liouville-Weyl and Liouville-Caputo general fractional derivative types with non-singular power-law kernel for the first time. The Fourier transforms and the anomalous diffusions are discussed in detail. The formulations are adopted to describe complex phenomena of the heat transfer problems.

scholarly journals In Depth Analysis of Stretch Function Resulting from Solving the Generalize Fractional-Order Bloch Equations Using Fractional Calculus

2020 ◽  
Vol 7 (4) ◽  
pp. 667-676
Author(s):  
Yahia Z. Rawash
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Exponential Decay ◽  
Anomalous Diffusion ◽  
Continuous Distribution ◽  
Bloch Equations ◽  
Diffusion Parameters ◽  
Depth Analysis ◽  
Magnetization Decay ◽  
Decay Profile

In this paper the stretch function resulting from solving the fractional-order Bloch equations using fractional calculus was discussed. This function has promising results to represent diffusion signal decay from MRI images. Conventional analyses of (DWI) measurements resolve the normalized magnetization decay profiles in terms of discrete and mono-exponential components with distinct lifetimes. In complex, heterogeneous biological and biophysical samples such as tissue, multi-exponential decay functions can appear to provide truer representation to normalized magnetization decay profile than the assumption of a mono-exponential decay, but the assumption of multiple discrete components is arbitrary and is often erroneous. Moreover, interactions, between both normalized magnetization and with their environment, can result in complex normalized magnetization decay profiles that represent a continuous distribution of lifetimes. The purpose in this paper is to study different factors that influence the stretch function strength, clarity, and contrast of MRI magnetization signal relaxation by manipulating the anomalous diffusion parameters Δ,δ,Gz,β and μ. of Bloch equations. Through this study, it was found that complex normalized magnetization decay profiles behave like stretch exponential function inside power law. Further developments of this study may be useful in optimizing anomalous diffusion in tissues with neurodegenerative, and ischemic diseases.

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