Abstract
Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.
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References
M. Ait Ichou, H. El Amri, A. Ezziani, On existence and uniqueness of solution for space-time fractional Zener model., Acta Appl. Math. (2020), to appear.
W. Arendt, C.J.K. Bety, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems Birkhäuser (2011).
T. Atanacković, M. Janev, S. Konjik, S. Pilipović, Complex fractional Zener model of wave propagation in ℝ. Fract. Calc. Appl. Anal. 21, No 5 (2018), 1313–1334; DOI: 10.1515/fca-2018-0069; https://www.degruyter.com/view/journals/fca/21/5/fca.21.issue-5.xml.
T. Atanacković, M. Janev, S. Konjik, S. Pilipović, Wave equation for generalized Zener model containing complex order fractional derivatives. Contin. Mech. Thermodyn. 29 (2017), 569–583.
T. Atanacković, M. Janev, S. Pilipović., Wave equation in fractional Zener-type viscoelastic media involving Caputo-Fabrizio fractional derivatives. Meccanica 54, No 1-2 (2019), 155–167.
T. Atanacković, M. Janev, S. Pilipović, On the restrictions in isothermal deformations of fractional Burgers model. Phil. Trans. R. Soc. A 378 (2020), No 20190278; DOI: 10.1098/rsta.2019.0278.
E. Bazhlekova, I. Bazhlekov, Transition from diffusion to wave propagation in fractional Jeffreys-type heat conduction equation. Fractal Fract. 4 (2020), No 32; DOI: 10.3390/fractalfract4030032.
M. Caputo, F. Mainardi, A new dissipation model based on memory mechanism., Pure Appl. Geophys. 91 (1971), 134–147.
M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids., Riv. Nuovo Cimento (Ser. II) 1 (1971), 161–198.
G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation Springer-Verlag, Berlin-Heidelberg (1974).
M. Fabrizio, Fractional rheological models for thermomechanical systems. Dissipation and free energies. Fract. Calc. Appl. Anal. 17, No 1 (2014), 206–233; DOI: 10.2478/s13540-014-0163-7; https://www.degruyter.com/view/journals/fca/17/1/fca.17.issue-1.xml.
I.M. Gelfand, N.Ya. Vilenkin, Generalized Functions - Vol 4: Applications of Harmonic Analysis Acad. Press, New York (1964).
R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications Springer Monographs in Mathematics, Springer, Heidelberg (2014), 2nd Ed. (2020).
N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45 (2006), 765–771.
R.G. Jaimez, M.J.V. Bonnet, On the Karhunen-Loéve expansion for transformed processes. Trabajos de Estadistica 2, No 2 (1987), 81–90.
S. Jaksi'c, B. Prangoski, Extension theorem of Whitney type for S’(Rd+) by use of the kernel theorem., Publ. de l'Institut Mathematique 99, No 113 (2016), 59–65.
S. Konjik, Lj. Oparnica, D. Zorica, Waves in fractional Zener type viscoelastic media., J. Math. Anal. Appl. 365 (2010), 259–268.
S. Konjik, Lj. Oparnica, D. Zorica, Waves in viscoelastic media described by a linear fractional model., Integral Transf. Spec Funct. 22 (2011), 283–291.
S. Konjik, Lj. Oparnica, D. Zorica, Distributed-order fractional constitutive stress-strain relation in wave propagation modeling., Z. Angew. Math. Phys. 70, No 51 (2019); DOI: 10.1007/s00033-019-1097-z.
P.H. Lam, H.C. So, Exponential sum approximation for Mittag-Leffler function and its application to fractional Zener wave equation., J. of Computational Physics 410 (2020), # 109389.
M. Loéve, Probability Theory II Springer Verlag (1978).
Y. Luchko, Fractional wave equation and damped waves., J. Math. Phys. 54 (2013), # 031505.
Y. Luchko, F. Mainardi, Y. Povstenko, Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation., Computers and Mathematics with Applications 66 (2013), 774–784.
F. Mainardi, Fractional Calculus and Waves in Linaear Viscoelasticity Imperial College, London (2010).
F. Mainardi, A historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 15, No 4 (2012), 712–717; DOI: 10.2478/s13540-012-0048-6; https://www.degruyter.com/view/journals/fca/15/4/fca.15.issue-4.xml.
S.P. Näsholm, S. Holm, On a fractional Zener elastic wave equation., Fract. Calc. Appl. Anal. 16, No 1 (2013), 26–50; DOI: 10.2478/s13540-013-0003-1; https://www.degruyter.com/view/journals/fca/16/1/fca.16.issue-1.xml.
B. Ø ksendal, Stochastic Differential Equations - An Introduction with Applications Springer-Verlag, Berlin-Heidelberg (2003).
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives - Theory and Applications Gordon and Breach Science Publ., Amsterdam (1993).
T. Soong, Random Differential Equations in Science and Engineering Academic Press (1973).
V.S. Vladimirov, Equations of Mathematical Physics Mir Publishers, Moscow (1984).
J.B. Walsh, An Introduction to Stochastic Partial Differential Equations Springer Lecture Notes in Mathematics (1980), 265–437.
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Atanacković, T., Pilipović, S. & Seleši, D. Wave Propagation Dynamics in a Fractional Zener Model with Stochastic Excitation. Fract Calc Appl Anal 23, 1570–1604 (2020). https://doi.org/10.1515/fca-2020-0079
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DOI: https://doi.org/10.1515/fca-2020-0079