Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 248175, 17 pages
http://dx.doi.org/10.1155/2012/248175
Research Article

Application of Integer and Fractional Models in Electrochemical Systems

Department of Electrical Engineering, Superior Institute of Engineering of Porto (ISEP), 4200-072 Porto, Portugal

Received 4 July 2011; Revised 7 October 2011; Accepted 8 October 2011

Academic Editor: Katica R. (Stevanovic) Hedrih

Copyright © 2012 Isabel S. Jesus and J. A. Tenreiro Machado. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. K. B. Oldham and J. Spanier, The Fractional Calculus, vol. 11 of Mathematics in Science and Engineering, Academic Press, London, UK, 1974. View at Zentralblatt MATH
  2. K. B. Oldham and J. Spanier, Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, Academic Press, London, UK, 1974. View at Zentralblatt MATH
  3. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. View at Zentralblatt MATH
  4. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at Zentralblatt MATH
  5. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. D. Monda and K. Biswas, “Electrical equivalent modeling of single component fractional order element with porous surface,” in Proceedings of the 5th International Conference on Sensor Technologies and Applications (SENSORCOMM '11), pp. 312–316, August 2011.
  7. D. Mondal and K. Biswas, “Performance study of fractional order integrator using single-component fractional order element,” IET Circuits, Devices & Systems, vol. 5, no. 4, pp. 334–342, 2011. View at Publisher · View at Google Scholar
  8. K. Biswas, Studies on design, development and performance analysis of capacitive type sensors, Ph.D. thesis, Indian Institute of Technology, Kharagpur, India, 2007.
  9. A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, vol. 378 of CISM Courses and Lectures, Springer, Vienna, Austria, 1997.
  10. L. Debnath, Fractional Calculus and Its Applications, 2002.
  11. J. A. T. Machado and I. S. Jesus, “A suggestion from the past?” FCAA—Journal of Fractional Calculus & Applied Analysis, vol. 7, no. 4, pp. 403–407, 2005. View at Google Scholar
  12. I. S. Jesus, J. A. T. Machado, J. B. Cunha, and M. F. Silva, “Fractional order electrical impedance of fruits and vegetables,” in Proceedings of the 25th IASTED International Conference on Modelling, Identification, and Control (MIC '06), pp. 489–494, January 2006. View at Scopus
  13. R. L. Magin, Fractional Calculus in Bioengineering, Begell House Connecticut, Paris, France, 2006.
  14. D. Baleanu, A. K. Golmankhaneh, and R. R. Nigmatullin, “Newtonian law with memory,” Nonlinear Dynamics, vol. 60, no. 1-2, pp. 81–86, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers & Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. C. M. Ionescu and R. D. Keyser, “Relations between fractional-order model parameters and lung pathology in chronic obstructive pulmonary disease,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 4, Article ID 4667643, pp. 978–987, 2009. View at Publisher · View at Google Scholar · View at Scopus
  17. K. R. S. Hedrih, “Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system),” Archive of Applied Mechanics, vol. 79, no. 6-7, pp. 529–540, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. H. Samavati, A. Hajimiri, A. R. Shahani, G. N. Nasserbakht, and T. H. Lee, “Fractal capacitors,” IEEE Journal of Solid-State Circuits, vol. 33, no. 12, pp. 2035–2041, 1998. View at Publisher · View at Google Scholar · View at Scopus
  19. A. K. Jonscher, Dielectric Relaxation in Solids, Chelesea Dielectric Press, London, UK, 1993.
  20. G. W. Bohannan, Analog Realization of a Fractional Order Control Element, Wavelength Electronics, Bozeman, Mo, USA, 2002.
  21. G. W. Bohannan, Interpretation of Complex Permittivity in Pure and Mixed Crystals, Wavelength Electronics.
  22. S. Westerlund and L. Ekstam, “Capacitor theory,” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 1, no. 5, pp. 826–839, 1994. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Westerlund, Dead Matter has Memory!, Causal Consulting, Kalmar, Sweden, 2002.
  24. E. Barsoukov and J. R. Macdonald, Impedance Spectroscopy, Theory, Experiment, and Applications, John Wiley & Sons, 2005.
  25. I. S. Jesus, J. A. T. Machado, and J. B. Cunha, “Fractional electrical dynamics in fruits and vegetables,” in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its Applications (FDA '06), pp. 374–379, 2006.
  26. I. S. Jesus, J. A. Tenreiro MacHado, and J. Boaventure Cunha, “Fractional electrical impedances in botanical elements,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1389–1402, 2008. View at Publisher · View at Google Scholar
  27. I. S. Jesus and J. A. T. MacHado, “Development of fractional order capacitors based on electrolyte processes,” Nonlinear Dynamics, vol. 56, no. 1-2, pp. 45–55, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  28. I. S. Jesus, J. A. T. Machado, and M. F. Silva, “Fractional order capacitors,” in Proceedings of the 27th IASTED International Conference on Modelling, Identification, and Control (MIC '08), February 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  29. I. S. Jesus, J. A. T. Machado, and R. S. Barbosa, “Fractional modelling of the electrical conduction in nacl electrolyte,” in Proceedings of the 6th EUROMECH Conference (ENOC '08), Saint Petersburg, RUSSIA, June 2008. View at Publisher · View at Google Scholar
  30. A. Le Méhauté, R. R. Nigmatullin, and L. Nivanen, Flèches du Temps et Géométrie Fractale, Hermès, Paris, France, 2nd edition, 1998.
  31. K. Falconer, Fractal Geometry—Mathematical Foundations and Application, John Wiley & Sons, England, UK, 1990.
  32. I. S. Jesus and J. A. T. Machado, “Comparing integer and fractional models in some electrical systems,” in Proceedings of the 4th IFAC Workshop on Fractional Differentiation and its Applications (FDA '10), Badajoz, Spain, October 2010.