Share This Article:

Fractional Order for Food Gums: Modeling and Simulation

Abstract Full-Text HTML XML Download Download as PDF (Size:190KB) PP. 305-309
DOI: 10.4236/am.2013.42046    4,734 Downloads   6,991 Views   Citations

ABSTRACT

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, i.e., 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 understand 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

S. David and A. Katayama, "Fractional Order for Food Gums: Modeling and Simulation," Applied Mathematics, Vol. 4 No. 2, 2013, pp. 305-309. doi: 10.4236/am.2013.42046.

References

[1] M. M. Luvielmo and A. R. P. Scamparini, “Goma Xantana: Producao, Recuperacao, Propriedades e Aplicacao,” Estudos Tecnologicos, Vol. 5, No. 1, 2009, pp. 50-67.
[2] R. B. Salah, S. Besbes, K. Chaari, A. Rhouma, H. Attia, C. Deroanne and C. Blecker, “Rheological and Physical Properties of Date Juice Palm By-Product (Phoenix dactylifera L.) and Commercial Xanthan Gums”, Journal of Texture Studies, Vol. 41, No. 2, 2010, pp. 125-138. doi:10.1111/j.1745-4603.2010.00217.x
[3] P. J. Whitcomb and C. W. Macosko, “Rheology of Xanthan Gum,” Journal of Rheology, Vol. 22, No. 5, 1978, pp. 493-505. doi:10.1122/1.549485
[4] J. A. Lopes da Silva, M. P. Gon?alves and M. A. Rao, “Viscoelastic Behavior of Mixtures of Locust Bean Gum and Pectin Dispersions,” Journal of Food Engineering, Vol. 18, No. 3, 1993, pp. 211-228. doi:10.1016/0260-8774(93)90087-Z
[5] M. Kobayashi and N. Nakahama, “Rheological Properties of Mixed Gels,” Journal of Texture Studies, Vol. 17, No. 2, 1986, pp. 161-174. doi:10.1111/j.1745-4603.1986.tb00402.x
[6] K. B. Oldham and J. Spanier, “The Fractional Calculus,” Academic Press, New York, 1974.
[7] S. A. David, J. L. Linares and E. M. J. A. Pallone, “Fractional Order Calculus: Historical Apologia, Basic Concepts and Some Applications,” Revista Brasileira de Ensino de Física, Vol. 33, No. 4, 2011, p. 4202. doi:10.1590/S1806-11172011000400002
[8] M. Caputo, “Distributed Order Differentia Equations Modeling Dieletric Induction and Diffusion,” Fractional Calculus and Applied Analysis, Vol. 4, No. 4, 2001, pp. 421-442.
[9] B. Ross, “Fractional Calculus,” Mathematics Magazine, Vol. 50, No. 3, 1977, pp. 115-122. doi:10.2307/2689497
[10] M. Caputo and F. Mainardi, “Linear Models of Dissipation in Anelastic Solids,” Rivista del Nuovo Cimento, Vol. 1, No. 2, 1971, pp. 161-198. doi:10.1007/BF02820620
[11] L. Ma and G. V. Barbosa-Canovas, “Simulating Viscoelastic Properties of Selected Food Gums and Gum Mixtures Using a Fractional Derivative Model,” Journal of Texture Studies, Vol. 27, No. 3, 2007, pp. 307-325. doi:10.1111/j.1745-4603.1996.tb00077.x
[12] K. W. Song, H. Y. Kuk and G. S. Chang, “Rheology of Concentrated Xanthan Gum Solutions: Oscillatory Shear Flow Behavior,” Korea-Australia Rheology Journal, Vol. 18, No. 2, 2006, pp. 67-81.
[13] R. L. Bagley, “Power Law and Fractional Calculus Model of Viscoelasticity,” AIAA Journal, Vol. 27, No. 10, 1989, pp. 1414-1417. doi:10.2514/3.10279

  
comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.