A detailed analysis for the fundamental solution of fractional vibration equation

Li-Li Liu; Jun-Sheng Duan

Open Mathematics (2015)

  • Volume: 13, Issue: 1
  • ISSN: 2391-5455

Abstract

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In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.

How to cite

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Li-Li Liu, and Jun-Sheng Duan. "A detailed analysis for the fundamental solution of fractional vibration equation." Open Mathematics 13.1 (2015): null. <http://eudml.org/doc/275840>.

@article{Li2015,
abstract = {In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.},
author = {Li-Li Liu, Jun-Sheng Duan},
journal = {Open Mathematics},
keywords = {Fractional derivatives and integrals; Fractional differential equations; Laplace transform; Fractional vibration},
language = {eng},
number = {1},
pages = {null},
title = {A detailed analysis for the fundamental solution of fractional vibration equation},
url = {http://eudml.org/doc/275840},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Li-Li Liu
AU - Jun-Sheng Duan
TI - A detailed analysis for the fundamental solution of fractional vibration equation
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - null
AB - In this paper, we investigate the solution of the fractional vibration equation, where the damping term is characterized by means of the Caputo fractional derivative with the order α satisfying 0 < α < 1 or 1 < α < 2. Detailed analysis for the fundamental solution y(t) is carried out through the Laplace transform and its complex inversion integral formula. We conclude that y(t) is ultimately positive, and ultimately decreases monotonically and approaches zero for the case of 0 < α < 1, while y(t) is ultimately negative, and ultimately increases monotonically and approaches zero for the case of 1 < α < 2. We also consider the number of zeros, the maximum zero and the maximum extreme point of the fundamental solution y(t) for specified values of the coefficients and fractional order.
LA - eng
KW - Fractional derivatives and integrals; Fractional differential equations; Laplace transform; Fractional vibration
UR - http://eudml.org/doc/275840
ER -

References

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  1. [1] B˘aleanu D., Diethelm K., Scalas E., Trujillo J.J., Fractional calculus models and numerical methods – Series on complexity, nonlinearity and chaos, World Scientific, Boston, 2012 
  2. [2] Gorenflo R., Mainardi F., Fractional calculus: integral and differential equations of fractional order. In: Carpinteri A., Mainardi F. (Eds.), Fractals and fractional calculus in continuum mechanics, Springer-Verlag, Wien/New York, 1997, pp. 223-276 
  3. [3] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006 Zbl1092.45003
  4. [4] Kiryakova V., Generalized fractional calculus and applications, Pitman Res. Notes in Math. Ser., Vol. 301, Longman Scientific & Technical and John Wiley & Sons, Inc., Harlow and New York, 1994 
  5. [5] Klafter J., Lim S.C., Metzler R., Fractional dynamics: recent advances, World Scientific, Singapore, 2011 Zbl1238.93005
  6. [6] Mainardi F., Fractional calculus and waves in linear viscoelasticity, Imperial College, London & World Scientific, Singapore, 2010 Zbl1210.26004
  7. [7] Miller K.S., Ross B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993 Zbl0789.26002
  8. [8] Oldham K.B., Spanier J., The fractional calculus, Academic Press, New York, 1974 Zbl0292.26011
  9. [9] Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999 
  10. [10] Rossikhin Y.A., Shitikova M.V., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanisms of solids, Appl. Mech. Rev., 1997, 50, 15-67 [Crossref] 
  11. [11] Koeller R.C., Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 1984, 51, 299-307 [Crossref] Zbl0544.73052
  12. [12] Xu M.Y., Tan W.C., Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solutions, Sci. China Ser. G, 2003, 46, 145-157 [Crossref] 
  13. [13] Chen W., An intuitive study of fractional derivative modeling and fractional quantum in soft matter, J. Vib. Control, 2008, 14, 1651-1657 [WoS][Crossref] Zbl1229.74009
  14. [14] Scott-Blair G.W., Analytical and integrative aspects of the stress-strain-time problem, J. Sci. Instr., 1944, 21, 80-84 
  15. [15] Scott-Blair G.W., A survey of general and applied rheology, Pitman, London, 1949 Zbl0033.23202
  16. [16] Bland D.R., The theory of linear viscoelasticity, Pergamon, Oxford, 1960 Zbl0108.38501
  17. [17] Bagley R.L., Torvik P.J., A generalized derivative model for an elastomer damper, Shock. Vib. Bull., 1979, 49, 135-143 
  18. [18] Beyer H., Kempfle S., Definition of physically consistent damping laws with fractional derivatives, ZAMM Z. Angew. Math. Mech., 1995, 75, 623-635 Zbl0865.70014
  19. [19] Achar B.N.N., Hanneken J.W., Clarke T., Response characteristics of a fractional oscillator, Physica A, 2002, 309, 275-288 Zbl0995.70017
  20. [20] Li M., Lim S.C., Chen S., Exact solution of impulse response to a class of fractional oscillators and its stability, Math. Probl. Eng., 2011, 2011, ID 657839 [WoS] Zbl1202.34018
  21. [21] Lim S.C., Li M., Teo L.P., Locally self-similar fractional oscillator processess, Fluct. Noise Lett., 2007, 7, L169-L179 [Crossref][WoS] 
  22. [22] Shen Y.J., Yang S.P., Xing H.J., Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative, Acta Phys. Sinica, 2012, 61, 110505-1-6 
  23. [23] Shen Y., Yang S., Xing H., Gao G., Primary resonance of Duffing oscillator with fractional-order derivative, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 3092-3100 [Crossref] Zbl1252.35274
  24. [24] Naber M., Linear fractionally damped oscillator, Int. J. Difference Equ., 2010, 2010, ID197020 Zbl1207.34010
  25. [25] Wang Z.H., Du M.L., Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system, Shock Vib., 2011, 18, 257-268 [WoS][Crossref] 
  26. [26] Li M., Lim S.C., Cattani C., Scalia M., Characteristic roots of a class of fractional oscillators, Adv. High Energy Phys., 2013, 2013, ID 853925 [WoS] Zbl1328.34007
  27. [27] Naranjani Y., Sardahi Y., Chen Y.Q., Sun J.Q., Multi-objective optimization of distributed-order fractional damping, Commun. Nonlinear Sci. Numer. Simul., 2015, 24, 159-168 [Crossref][WoS] 
  28. [28] Li M., Li Y.C., Leng J.X., Power-type functions of prediction error of sea level time series, Entropy, 2015, 17, 4809-4837 
  29. [29] Li C.P., Deng W.H., Xu D., Chaos synchronization of the Chua system with a fractional order, Physica A, 2006, 360, 171-185 
  30. [30] Zhang W., Liao S.K., Shimizu N., Dynamic behaviors of nonlinear fractional-order differential oscillator, J. Mech. Sci. Tech., 2009, 23, 1058-1064 [Crossref] 
  31. [31] Wang Z.H., Hu H.Y., Stability of a linear oscillator with damping force of the fractional-order derivative, Sci. China Ser. G, 2010, 53, 345-352 [WoS][Crossref] 
  32. [32] Li C., Ma Y., Fractional dynamical system and its linearization theorem, Nonlinear Dynam., 2013, 71, 621-633 [WoS] Zbl1268.34019
  33. [33] Yang X.J., Baleanu D., Srivastava H.M., Local fractional similarity solution for the diffusion equation defined on Cantor sets, Appl. Math. Lett., 2015, 47, 54-60 [Crossref] Zbl06477315
  34. [34] Yang X.J., Srivastava H.M., An asymptotic perturbation solution for a linear oscillator of free damped vibrations in fractal medium described by local fractional derivatives, Commun. Nonlinear Sci. Numer. Simul., 2015, 29, 499-504 [Crossref][WoS] 
  35. [35] Yang X.J., Srivastava H.M., Cattani C., Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 2015, 67, 752-761 
  36. [36] Yang X.J., Baleanu D., Baleanu M.C., Observing diffusion problems defined on cantor sets in different coordinate systems, Therm. Sci., 2015, 19(S1), 151-156 [WoS][Crossref] 
  37. [37] Yan S.P., Local fractional Laplace series expansion method for diffusion equation arising in fractal heat transfer, Therm. Sci., 2015, 19, S131-S135 [WoS][Crossref] 
  38. [38] Duan J.S., Time- and space-fractional partial differential equations, J. Math. Phys., 2005, 46, 13504-13511 [Crossref] 
  39. [39] Duan J.S., Fu S.Z., Wang Z., Fractional diffusion-wave equations on finite interval by Laplace transform, Integral Transforms Spec. Funct., 2014, 25, 220-229 [WoS] Zbl1297.35270
  40. [40] Davies B., Integral transforms and their applications, 3rd ed., Springer-Verlag, New York, 2001 Zbl0996.44001
  41. [41] Duan J.S., Wang Z., Fu S.Z., The zeros of the solutions of the fractional osciliation equation, Fract. Calc. Appl. Anal., 2014, 17, 10-12 [Crossref] 
  42. [42] Duan J.S., Wang Z., Liu Y.L., Qiu X., Eigenvalue problems for fractional ordinary differential equations, Chaos Solitons Fractals, 2013, 46, 46-53 [Crossref][WoS] Zbl1258.34041

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