zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A note on fractional-order derivatives of periodic functions. (English) Zbl 1191.93062
Summary: It is shown that the fractional-order derivatives of a periodic function with a specific period cannot be a periodic function with the same period. The fractional-order derivative considered here can be obtained based on each of the well-known definitions Grunwald-Letnikov definition, Riemann-Liouville definition and Caputo definition. This concluded point confirms the result of a recently published work proving the non-existence of periodic solutions in a class of fractional-order models. Also, based on this point it can be easily proved the absence of periodic responses in a wider class of fractional-order models. Finally, some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional-order systems.

93C15Control systems governed by ODE
34A08Fractional differential equations
34K13Periodic solutions of functional differential equations
Full Text: DOI
[1] Aoun, M.; Malti, R.; Levron, F.; Oustaloup, A.: Synthesis of fractional Laguerre basis for system approximation, Automatica 43, 1640-1648 (2007) · Zbl 1128.93019 · doi:10.1016/j.automatica.2007.02.013
[2] Diethelm, K.; Ford, N. J.: Multi-order fractional differential equations and their numerical solution, Applied mathematics and computation 154, 621-640 (2004) · Zbl 1060.65070 · doi:10.1016/S0096-3003(03)00739-2
[3] Ding, Y.; Ye, H.: A fractional-order differential equation model of HIV infection of CD4+ T-cells, Mathematical and computer modelling 50, 386-392 (2009) · Zbl 1185.34005 · doi:10.1016/j.mcm.2009.04.019
[4] Edwards, J. T.; Ford, N. J.; Simpson, A. C.: The numerical solution of linear multi-term fractional differential equations: systems of equations, Journal of computational and applied mathematics 148, 401-418 (2002) · Zbl 1019.65048 · doi:10.1016/S0377-0427(02)00558-7
[5] Feliu-Batlle, V.; Pérez, R. Rivas; García, F. J. Castillo; Rodriguez, L. Sanchez: Smith predictor based robust fractional-order control: application to water distribution in a Main irrigation canal pool, Journal of process control 19, No. 3, 506-519 (2009)
[6] Fitt, A. D.; Goodwin, A. R. H.; Ronaldson, K. A.; Wakeham, W. A.: A fractional differential equation for a MEMS viscometer used in the oil industry, Journal of computational and applied mathematics 229, 373-381 (2009) · Zbl 1235.34201
[7] Hartley, T. T.; Lorenzo, C. F.: Fractional-order system identification based on continuous order-distributions, Signal processing 83, 2287-2300 (2003) · Zbl 1145.93433 · doi:10.1016/S0165-1684(03)00182-8
[8] Li, Y.; Chen, Y. Q.; Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica 45, 1965-1969 (2009) · Zbl 1185.93062 · doi:10.1016/j.automatica.2009.04.003
[9] Li, C.; Deng, W.: Remarks on fractional derivatives, Applied mathematics and computation 187, 777-784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163
[10] Monje, C. A.; Vinagre, B. M.; Feliu, V.; Chen, Y. Q.: Tuning and auto-tuning of fractional order controllers for industry applications, Control engineering practice 16, 798-812 (2008)
[11] Nimmo, S.; Evans, A. K.: The effects of continuously varying the fractional differential order of a chaotic nonlinear system, Chaos, solitons and fractals 10, 1111-1118 (1999) · Zbl 0980.34032 · doi:10.1016/S0960-0779(98)00088-5
[12] Oldham, K. B.: Fractional differential equations in electrochemistry, Advances in engineering software 41, 9-12 (2010) · Zbl 1177.78041 · doi:10.1016/j.advengsoft.2008.12.012
[13] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[14] Poinot, T.; Trigeassou, J. C.: Identification of fractional systems using an output-error technique, Nonlinear dynamics 38, 133-154 (2004) · Zbl 1134.93324 · doi:10.1007/s11071-004-3751-y
[15] Reyes-Melo, M. E.; González-González, V. A.; Guerrero-Salazar, C. A.; García-Cavazos, F.; Ortiz-Méndez, U.: Application of fractional calculus to the modeling of the complex rheological behavior of polymers: from the Glass transition to flow behavior. I. the theoretical model, Journal of applied polymer science 108, No. 2, 731-737 (2008)
[16] Tavazoei, M. S.; Haeri, M.: A proof for non existence of periodic solutions in time invariant fractional order systems, Automatica 45, 1886-1890 (2009) · Zbl 1193.34006 · doi:10.1016/j.automatica.2009.04.001
[17] Tavazoei, M. S.; Haeri, M.: Describing function based methods for predicting chaos in a class of fractional order differential equations, Nonlinear dynamics 57, 363-373 (2009) · Zbl 1176.34051 · doi:10.1007/s11071-008-9447-y
[18] Tavazoei, M. S.; Haeri, M.; Attari, M.; Bolouki, S.; Siami, M.: More details on analysis of fractional-order van der Pol oscillator, Journal of vibration and control 15, No. 6, 803-819 (2009) · Zbl 1273.70037
[19] Tavazoei, M. S.; Haeri, M.; Jafari, S.; Bolouki, S.; Siami, M.: Some applications of fractional calculus in suppression of chaotic oscillations, IEEE transactions on industrial electronics 11, 4094-4101 (2008)
[20] Tavazoei, M. S.; Haeri, M.; Nazari, N.: Analysis of undamped oscillations generated by marginally stable fractional order systems, Signal processing 88, 2971-2978 (2008) · Zbl 1151.94415 · doi:10.1016/j.sigpro.2008.07.002