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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]:
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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2011, Issue 1(22), Pages 255–268 (Mi vsgtu932)  

This article is cited in 16 scientific papers (total in 16 papers)

Procedings of the 2nd International Conference "Mathematical Physics and its Applications"
Mechanics

Rheological model of viscoelastic body with memory and differential equations of fractional oscillator

E. N. Ogorodnikov, V. P. Radchenko, N. S. Yashagin

Dept. of Applied Mathematics and Computer Science, Samara State Technical University, Samara

Abstract: One-dimensional generalized rheologic model of viscoelastic body with Riemann-Liouville derivatives is considered. Instead of derivatives of order $\alpha>1$ there are employed in defining relations derivatives of order $0<\alpha<1$ from integer derivatives. Its shown, that the differential equation for the deformation with given dependence of the tension from the time with classical initial conditions of Cauchy are reduced to the Volterra integral equations. Some variants of the generalized fractional Voigts model are considered. Explicit solutions for corresponding differential equation for the deformation are found out. Its indicated, that these solutions coincide with the classical ones when the fractional parameter vanishes.

Keywords: rheological model of viscoelastic body, differential equations with fractional Riemann–Liouville derivatives Mittag–Leffler type special functions

DOI: https://doi.org/10.14498/vsgtu932

Full text: PDF file (605 kB) (published under the terms of the Creative Commons Attribution 4.0 International License)
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Bibliographic databases:

Document Type: Article
UDC: 539.313:517.968.72
MSC: Primary 74D10; Secondary 26A33
Original article submitted 12/XII/2010
revision submitted – 17/II/2011

Citation: E. N. Ogorodnikov, V. P. Radchenko, N. S. Yashagin, “Rheological model of viscoelastic body with memory and differential equations of fractional oscillator”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 1(22) (2011), 255–268

Citation in format AMSBIB
\Bibitem{OgoRadYas11}
\by E.~N.~Ogorodnikov, V.~P.~Radchenko, N.~S.~Yashagin
\paper Rheological model of viscoelastic body with memory and differential equations of fractional oscillator
\jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]
\yr 2011
\vol 1(22)
\pages 255--268
\mathnet{http://mi.mathnet.ru/vsgtu932}
\crossref{https://doi.org/10.14498/vsgtu932}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. N. Ogorodnikov, N. S. Yashagin, “Metody resheniya nachalno-kraevykh zadach dlya differentsialnykh uravnenii s volnovym operatorom i mladshimi drobnymi proizvodnymi Rimana-Liuvillya”, Trudy vosmoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem, Matematicheskoe modelirovanie i kraevye zadachi, SamGTU, Samara, 2011, 140–145  elib
    2. E. N. Ogorodnikov, “Ob odnom klasse drobnykh differentsialnykh uravnenii matematicheskikh modelei dinamicheskikh sistem s pamyatyu”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(30) (2013), 245–252  mathnet  crossref
    3. E. N. Ogorodnikov, “Postanovka i reshenie zadachi tipa koshi dlya odnogo klassa modelnykh dinamicheskikh sistem s pamyatyu”, Trudy devyatoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem, Matematicheskoe modelirovanie i kraevye zadachi, SamGTU, Samara, 2013, 147–152  elib
    4. E. N. Ogorodnikov, “Nekotorye reologicheskie modeli vyazkouprugogo tela s pamyatyu i sootvetstvuyuschie im drobnoostsilyatsionnye uravneniya”, Trudy devyatoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem, Matematicheskoe modelirovanie i kraevye zadachi, SamGTU, Samara, 2013, 49–53  elib
    5. L. G. Abusaitova, E. N. Ogorodnikov, “Sravnitelnyi analiz drobnykh reologicheskikh modelei kelvina i zenera, osnovannykh na ispolzovanii apparata integro-differentsirovaniya Rimana-Liuvillya”, Trudy devyatoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem, Matematicheskoe modelirovanie i kraevye zadachi, SamGTU, Samara, 2013, 12–15  elib
    6. V. A. Kubyshkin, S. S. Postnov, Drobnoe integro-differentsialnoe ischislenie i ego prilozheniya v teorii upravleniya, v. 2, Podkhody k interpretatsii drobnykh operatsii. Dinamicheskie sistemy drobnogo poryadka, Institut problem upravleniya im. V. A. Trapeznikova RAN, Moskva, 2013, 73 pp.  elib
    7. A. S. Ovsienko, “Razrabotka metodov identifikatsii parametrov differentsialnykh uravnenii s drobnoi proizvodnoi Rimana–Liuvillya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 1(34) (2014), 134–144  mathnet  crossref  zmath  elib
    8. E. N. Ogorodnikov, L. G. Abusaitova, “Opredelyayuschie sootnosheniya i nachalnye zadachi dlya vyazkouprugikh sred s drobnymi operatorami RimanaLiuvillya”, Materialy VIII Vserossiiskoi konferentsii po mekhanike deformiruemogo tverdogo tela. Ch. 2 (Cheboksary, 1621 iyunya 2014 g.), eds. N. F. Morozov, B. G. Mironov, A. V. Manzhirov, Chuvash. gos. ped. un-t, Cheboksary, 2014, 105–107
    9. V. P. Radchenko, E. N. Ogorodnikov, L. G. Abusaitova, “Matematicheskoe modelirovanie polzuchesti na osnove apparata drobnogo integro-differentsirovaniya Rimana-Liuvillya”, Deformirovanie i razrushenie strukturno-neodnorodnykh sred i konstruktsii, Sbornik materialov III Vserossiiskoi konferentsii, posvyaschennoi 100-letiyu so dnya rozhdeniya akademika Yu.N. Rabotnova, Novosibirskii gosudarstvennyi tekhnicheskii universitet, Novosibirsk, 2014, 88  elib
    10. E. N. Ogorodnikov, “Ob odnoi matematicheskoi modeli deformirovaniya reologicheskikh sred s pamyatyu”, Uravneniya smeshannogo tipa, rodstvennye problemy analiza i informatiki, Tretii Mezhdunarodnyi Rossiisko-Kazakhskii simpozium, Redaktsiya zhurnala Elbrus, Nalchik, 2014, 156–158  elib
    11. E. N. Ogorodnikov, V. P. Radchenko, L. G. Ungarova, “Matematicheskoe modelirovanie nasledstvenno uprugogo deformiruemogo tela na osnove strukturnykh modelei i apparata drobnogo integro-differentsirovaniya Rimana–Liuvillya”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:1 (2016), 167–194  mathnet  crossref  zmath  elib
    12. V. P. Orlov, D. A. Rode, M. A. Pliev, “Weak solvability of the generalized Voigt viscoelasticity model”, Siberian Math. J., 58:5 (2017), 859–874  mathnet  crossref  crossref  isi  elib  elib
    13. E. N. Ogorodnikov, V. P. Radchenko, L. G. Ungarova, “Mathematical models of nonlinear viscoelasticity with operators of fractional integro-differentiation”, PNRPU Mechanics Bulletin, 2018, no. 2, 147–161 (In Russian)  crossref  scopus
    14. A. V. Khokhlov, “Analiz svoistv krivykh polzuchesti s proizvolnoi nachalnoi stadiei nagruzheniya, porozhdaemykh lineinoi teoriei nasledstvennosti”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 22:1 (2018), 65–95  mathnet  crossref
    15. V. Zvyagin, V. Orlov, “Weak solvability of fractional voigt model of viscoelasticity”, Discrete and Continuous Dynamical Systems- Series A, 38:12 (2018), 6327–6350  crossref  isi  scopus
    16. V. G. Zvyagin, V. P. Orlov, “On solvability of an initial-boundary value problem for a viscoelasticity model with fractional derivatives”, Siberian Math. J., 59:6 (2018), 1073–1089  mathnet  crossref  crossref  isi
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