A physical interpretation of fractional calculus in observables terms: analysis of the fractional time constant and the transitory response

J.F. Gómez-Aguilar^a,*, R. Razo-Hernández^b, and D. Granados-Lieberman^b

ª Departamento de Ingeniería Eléctrica, División de Ingenierías Campus Irapuato-Salamanca, Universidad de Guanajuato Carretera Salamanca-Valle de Santiago, km. 3.5 + 1.8 km Comunidad de Palo Blanco, Salamanca Guanajuato. México, * Tel: (464) 6479940 e-mail: jgomez@ugto.mx

^b Departamento de Electromecánica, Instituto Superior de Irapuato, Irapuato, Guanajuato, México, e-mail: jorazo@itesi.edu.mx, david.granados@itesi.edu.mx

Abstract

This work presents the analysis of the fractional time constant and the transitory response (delay, rise, and settling times) of a RC circuit as a physical interpretation of fractional calculus in observables terms, the definition of Caputo fractional derivative is applied. The physical interpretation of these observables allows a clearer understanding of the concept of fractional derivative.

Keywords: Fractional calculus; fractional time constant; fractional differential equations; transitory response.

Resumen

Este trabajo presenta el análisis de la constante de tiempo transitoria y de la respuesta en frecuencia (tiempo de retraso, elevación y asentamiento) de un circuito RC como una interpretación física del cálculo fraccionario en términos de estos observables, la definición de derivada fraccionaria de Caputo es aplicada. La interpretación física de estos observables permite tener un entendimiento claro del concepto de derivada fraccionaria.

Descriptores: Calculo fraccionario; constante de tiempo fraccionaria; ecuaciones diferenciales fraccionarias; respuesta transitoria.

PACS: 03.50.De; 45.10.Hj; 05.45.-a

DESCARGAR ARTÍCULO EN FORMATO PDF

References

1. K.B. Oldham and J. Spanier, The Fractional Calculus. (Academic Press, New York, 1974).         [ Links ]

2. K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. (John Wiley, NY, 1993).         [ Links ]

3. M. Duarte Ortiguera, Fractional Calculus for Scientists and Engineers (Springer, NY, 2011).         [ Links ]

4. D. Cafagna, Fractional Calculus: A mathematical Tool From Past for Present Engineers. (IEEE Industrial Electronics Magazine, Summer 2007).         [ Links ]

5. S.G. Samko, A.A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Theoryand Applications. (Langhorne, PA: Gordon and Breach Science Publishers, 1993).         [ Links ]

6. I. Podlubny, Fractional Differential Equations (Academic Press, New York, 1999).         [ Links ]

7. D. Baleanu, Fractional Calculus Models and Numerical Methods (World Scientific Publishing Company, 2012).         [ Links ]

8. C.A. Monje, Y.Q. Chen, B.M. Vinagre, D. Xue, and V. Feliu, Fractional-Order Systems and Controls, (Series: Advances in Industrial Control, Springer, 2010).         [ Links ]

9. R. Caponetto, G. Dongola, L. Fortuna, and I. Petras, Fractional Order Systems: Modeling and Control Applications, (Singapore: World Scientific, 2010).         [ Links ]

10. D. Baleanu, Z.B. Günvenc, and J.A. Tenreiro Machado, (Eds). New Trends in Nanotechnologyand Fractional Calculus Applications. (Springer, 2010).         [ Links ]

11. D. Baleanu, K. Diethelm, E. Scalas, and J.J. Trujillo, Fractional Calculus Models and Numerical Methods. (Series on Complexity, Nonlinearity and Chaos. World Scientific, 2012).         [ Links ]

12. D. Baleanu, A. K. Golmankhaneh, A. K. Golmankhaneh, and R. R. Nigmatullin, Nonlinear Dynamics 60 (2010) 81-86.         [ Links ]

13. W. Wyss, J. Math. Phys. 27 (1986) 2782-2785.         [ Links ]

14. R. Hilfer, J. Phys. Chem. 104 (2000) 3914-3917.         [ Links ]

15. R. Metzler, and J. Klafter, Phys. Rep. 339, (2000) 1-77.         [ Links ]

16. R. Metzler, and J. Klafter, J. Phys. A37 (2004) R161-R208.         [ Links ]

17. O.P. Agrawal, J.A. Tenreiro-Machado, and I. Sabatier, (Eds.) Fractional Derivatives and Their Applications: Nonlinear Dynamics, 38, (Berlin: Springer-Verlag, 2004).         [ Links ]

18. R. Hilfer, (Ed.) Applications of Fractional Calculus in Physics (Singapore: World Scientific, 2000).         [ Links ]

19. S. Das, Functional Fractional Calculus for System Identification and Controls (Springer, NY, 2008).         [ Links ]

20. B.J. West, M. Bologna, and P. Grigolini, Physics of Fractional Operators, (Berlin: Springer-Verlag, 2003).         [ Links ]

21. R.L. Magin, Fractional Calculus in Bioengineering (Roddin: Begell House Publisher, 2006).         [ Links ]

22. M. Caputo and F. Mainardi, Pure and App. Geo. 91 (1971) 134-147.         [ Links ]

23. S. Westerlund, Causality, (University of Kalmar, Rep. No. 940426, 1994).         [ Links ]

24. F. Riewe, Physical ReviewE 53 (1996) 1890-1899.         [ Links ]

25. M.A.E. Herzallah, I. Muslih Sami, Dumitru Baleanu, and M. Rabei Eqab, Nonlinear Dynamics 66 (2011) 549-555.         [ Links ]

26. A. Khalili Golmankhaneh, M. Yengejeh Ali, and D. Baleanu, International Journal of Theoretical Physics 51 (2012) 2909-2916.         [ Links ]

27. Alireza K. Golmankhaneh, and Lap Lambert, Investigations in Dynamics: With Focus on Fractional Dynamics (Academic Publishing, 2012).         [ Links ]

28. I. Muslih Sami, M. Saddallah, D. Baleanu, and E. Rabei, Romanian Journal of Physics 55 (2010) 659-663.         [ Links ]

29. D. Baleanu, I. Muslih Sami, and M. Rabei Eqab, Nonlinear Dynamics 53 (2008) 67-74.         [ Links ]

30. V. Uchaikin, Fractional Calculus. (Edit. Artishok. In Russian, 2008).         [ Links ]

31. B. Mandelbrot, The Fractal Geometry of Nature, (San Francisco, CA: Freeman, 1982).         [ Links ]

32. K. Alireza, V. F. Golmankhaneh, and D. Baleanu, Romanian Reports in Physics 65 (2013) 84-93.         [ Links ]

33. I. Petrás, I. Podlubny, P. O'Leary, L. Dorcák, B.M. Vinagre, Analoge Realization of Fractional-Order Controllers. (Tu Kosice: BERG, Faculty, Slovakia, 2002).         [ Links ]

34. I. Petras. Fractional-Order Memristor-Based Chua's Circuit IEEE Transactions on Circuits and Systems-II: Express Briefs, 57, Issue. 12, 975-979, (2010).         [ Links ]

35. I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation, (London: Springer and Beijing: HEP, 2011).         [ Links ]

36. N. Engheta, Antennas and Propagation Magazine 39 (1997) 35-46.         [ Links ]

37. A. S. Balankin, B. Mena, J. Patino, and D. Morales, Physics Letters A 377 (2013) 738-788.         [ Links ]

38. V. E. Tarasov, and J. J. Trujillo, Annals of Physics 334 (2013) 1-23.         [ Links ]

39. V.E. Tarasov, Physics of Plasmas 12 (2005).         [ Links ]

40. V.E. Tarasov, Modern Physics Letters A 21 (2006) 1587-1600.         [ Links ]

41. V.E. Tarasov, Fractional Dynamics. (Springer. HEP. 2011).         [ Links ]

42. H. Nasrolahpour, Comm. Nonl. Sci. Num. Simul. 18 (2013) 2589-2593.         [ Links ]

43. J.F. Gómez-Aguilar, J.J. Rosales-García, J.J. Bernal-Alvarado, T. Córdova-Fraga, and R. Guzmán-Cabrera, Rev. Mex. Fís 58 (2012) 348-352.         [ Links ]

44. J.J. Rosales, M. Guía, J.F. Gómez, and V.I. Tkach, Discontinuity, Nonlinearityand Complexity 1 (2012) 325-335.         [ Links ]

45. I. Podlubny, Fract. Calc. App. Anal. 5 (2002) 367-386.         [ Links ]

46. M. Moshre-Torbati and J.K. Hammond, J. Franklin Inst. 335B (1998) 1077-1086.         [ Links ]

47. W. Bolton, Ingeniería de Control, 2 ed., Alfaomega, (2001).         [ Links ]