Fractional Order for Food Gums: Modeling and Simulation

Abstract

<span style="font-family:Verdana;"></span> <p align="justify"> <span><span style="font-family:;">Fractional order calculus can represent systems with high-order dynamics and complex nonlinear phenomena using few coefficients, since the arbitrary order of the derivatives provides an additional degree of freedom to fit a specific behavior. Numerous mathematicians have contributed to the history of fractional calculus by attempting to solve a fundamental problem to the best of their understanding. Each researcher sought a definition and therefore different approaches, which has led to various definitions of differentiation and anti-differentiation of non-integer orders that are proven equivalent. Although all these definitions may be equivalent, from one specific standpoint, </span><i><span style="font-family:;">i.e.</span></i><span style="font-family:;">, for a specific application, some definitions seem more attractive. Furthermore, it is well known that food gums are complex carbohydrates that can suit for a wide variety of functions in the context of food engineering. The viscoelastic behavior of food gums is crucial for these applications and formulations of new or improved food products. Small progress has been made to un</span></span><span style="font-family:;">derstand the viscoelastic behavior of food gums and there are few studies in the literature about these models. In this paper, we applied the Riemann-Liouville approach and the Fourier transform in order to obtain numerical simulations results of a fractional derivative model based on previous literature that to make a quantitative description of the viscoelastic properties behavior for a food gum. The results reveal that the fractional model shows good simulation capability and can be an attractive means for predicting and to elucidate the dynamic viscoelastic behavior of food gums.</span> </p

The application of fractional derivatives through Riemannliouville approach to xanthan gum viscoelasticity

“…The following is an experimental data table with a simplified parameter model derived from the study in [15], [5], where G' is storage modulus and G" is loss modulus, and k1, k2, 1  , and 2  are simplified parameters.…”

“…With  is the shear stress,  is the rate of shear strain,  is the viscosity of the fluid, and t is time. Linear elasticity is defined by Hooke's Law [5], that is:…”

“…Then the least squares method was used to determine the linear fractional linear model of fractional viscolysis parameters from storage measurement and modulus loss. Further studies [5], still with the Riemann-Liouville fractional derivative model to investigate viscoelasticity in xanthan gum, with 0.5%, 1.0%, 2.0%, and 4.0%.…”

Vectorial Iterative Fractional Laplace Transform Method for the Analytic Solutions of Fractional Cauchy-Riemann Systems Partial Differential Equations

“…Fractional calculus is also applicable to problems in: polymer science, polymer physics, biophysics, rheology, and thermodynamics [6]. In addition, it is applicable to problems in: electrochemical process [2,3,4], control theory [4,13], physics [14], science and engineering [8], transport in semi-infinite medium [3], signal processing [15], food science [16], food gums [17], fractional dynamics [18,19], modeling Cardiac tissue-electrode interface [20], food engineering and econophysics [13], complex dynamics in biological tissues [21], viscoelasticity [4,14,16,22,10], modeling oscillation systems [23]. Some of these mentioned applications were tried to be touched as follows.…”

Analytic solutions of time fractional diffusion equations by fractional reduced differential transform method (FRDTM)

“…It is also applicable to problems in polymer science, polymer physics, biophysics, rheology, and thermodynamics (Hilfer, 2000). In addition, it is applicable to problems in: electrochemical process (Millar and Ross, 1993;Oldham and Spanier, 1974;Podlubny, 1999), control theory (David et al, 2011;Podlubny, 1999), physics (Sabatier et al, 2007), science and engineering (Kumar and Saxena, 2016), transport in semi-infinite medium (Oldham and Spanier, 1974), signal processing (Sheng et al, 2011), self-similar protein dynamics (Glockle and Nonnenmacher, 1995), food science (Rahimy, 2010), food gums (David and Katayama, 2013), fractional dynamics (Tarsov, 2011;Zaslavsky, 2005), quantum dynamics (Iomin, 2009), modeling cardiac tissue electrode interface (Magin, 2008), food engineering and econophysics (David et al, 2011), Hamiltonian chaotic systems (Hilfer, 2000;Zaslavsky, 2005), complex dynamics in biological tissues (Margin, 2010), viscoelasticity (Dalir and Bashour, 2010;Mainardi, 2010;Podlubny, 1999;Rahimy, 2010;Sabatier et al, 2007), control science (Shanantu Sabatier et al, 2007), quantum mechanics (Herrmann, 2011), modeling Kenea 15 oscillation systems (Gomez-Aguilar et al, 2015). Some of these mentioned applications were tried to be touched as follows.…”