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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access November 26, 2015

A detailed analysis for the fundamental solution of fractional vibration equation

  • Li-Li Liu and Jun-Sheng Duan
From the journal Open Mathematics


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.


[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 10.1142/8180Search in Google Scholar

[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 Search in Google Scholar

[3] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006 Search in Google Scholar

[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 Search in Google Scholar

[5] Klafter J., Lim S.C., Metzler R., Fractional dynamics: recent advances, World Scientific, Singapore, 2011 10.1142/8087Search in Google Scholar

[6] Mainardi F., Fractional calculus and waves in linear viscoelasticity, Imperial College, London & World Scientific, Singapore, 2010 10.1142/p614Search in Google Scholar

[7] Miller K.S., Ross B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993 Search in Google Scholar

[8] Oldham K.B., Spanier J., The fractional calculus, Academic Press, New York, 1974 Search in Google Scholar

[9] Podlubny I., Fractional differential equations, Academic Press, San Diego, 1999 Search in Google Scholar

[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 10.1115/1.3101682Search in Google Scholar

[11] Koeller R.C., Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 1984, 51, 299-307 10.1115/1.3167616Search in Google Scholar

[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 10.1360/03yg9020Search in Google Scholar

[13] Chen W., An intuitive study of fractional derivative modeling and fractional quantum in soft matter, J. Vib. Control, 2008, 14, 1651-1657 10.1177/1077546307087398Search in Google Scholar

[14] Scott-Blair G.W., Analytical and integrative aspects of the stress-strain-time problem, J. Sci. Instr., 1944, 21, 80-84 10.1088/0950-7671/21/5/302Search in Google Scholar

[15] Scott-Blair G.W., A survey of general and applied rheology, Pitman, London, 1949 Search in Google Scholar

[16] Bland D.R., The theory of linear viscoelasticity, Pergamon, Oxford, 1960 Search in Google Scholar

[17] Bagley R.L., Torvik P.J., A generalized derivative model for an elastomer damper, Shock. Vib. Bull., 1979, 49, 135-143 Search in Google Scholar

[18] Beyer H., Kempfle S., Definition of physically consistent damping laws with fractional derivatives, ZAMM Z. Angew. Math. Mech., 1995, 75, 623-635 10.1002/zamm.19950750820Search in Google Scholar

[19] Achar B.N.N., Hanneken J.W., Clarke T., Response characteristics of a fractional oscillator, Physica A, 2002, 309, 275-288 10.1016/S0378-4371(02)00609-XSearch in Google Scholar

[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 10.1155/2011/657839Search in Google Scholar

[21] Lim S.C., Li M., Teo L.P., Locally self-similar fractional oscillator processess, Fluct. Noise Lett., 2007, 7, L169-L179 10.1142/S0219477507003817Search in Google Scholar

[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 10.7498/aps.61.110505Search in Google Scholar

[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 10.1016/j.cnsns.2011.11.024Search in Google Scholar

[24] Naber M., Linear fractionally damped oscillator, Int. J. Difference Equ., 2010, 2010, ID197020 10.1155/2010/197020Search in Google Scholar

[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 10.1155/2011/253130Search in Google Scholar

[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 10.1155/2013/853925Search in Google Scholar

[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 10.1016/j.cnsns.2014.12.011Search in Google Scholar

[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 10.3390/e17074809Search in Google Scholar

[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 10.1016/j.physa.2005.06.078Search in Google Scholar

[30] Zhang W., Liao S.K., Shimizu N., Dynamic behaviors of nonlinear fractional-order differential oscillator, J. Mech. Sci. Tech., 2009, 23, 1058-1064 10.1007/s12206-009-0341-4Search in Google Scholar

[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 10.1007/s11433-009-0291-ySearch in Google Scholar

[32] Li C., Ma Y., Fractional dynamical system and its linearization theorem, Nonlinear Dynam., 2013, 71, 621-633 10.1007/s11071-012-0601-1Search in Google Scholar

[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 10.1016/j.aml.2015.02.024Search in Google Scholar

[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 10.1016/j.cnsns.2015.06.006Search in Google Scholar

[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 Search in Google Scholar

[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 10.2298/TSCI141126065YSearch in Google Scholar

[37] Yan S.P., Local fractional Laplace series expansion method for diffusion equation arising in fractal heat transfer, Therm. Sci., 2015, 19, S131-S135 10.2298/TSCI141010063YSearch in Google Scholar

[38] Duan J.S., Time- and space-fractional partial differential equations, J. Math. Phys., 2005, 46, 13504-13511 10.1063/1.1819524Search in Google Scholar

[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 10.1080/10652469.2013.838759Search in Google Scholar

[40] Davies B., Integral transforms and their applications, 3rd ed., Springer-Verlag, New York, 2001 Search in Google Scholar

[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 10.2478/s13540-014-0152-xSearch in Google Scholar

[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 10.1016/j.chaos.2012.11.004Search in Google Scholar

Received: 2015-9-13
Accepted: 2015-10-21
Published Online: 2015-11-26

©2015 Li-Li Liu, Jun-Sheng Duan

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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