Abstract
We study the heat conduction with a general form of a constitutive equation containing fractional derivatives of real and complex order. Using the entropy inequality in a weak form, we derive sufficient conditions on the coefficients of a constitutive equation that guarantee that the second law of thermodynamics is satisfied. This equation, in special cases, reduces to known ones. Moreover, we present a solution of a temperature distribution problem in a semi-infinite rod with the proposed constitutive equation.
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Atanackovic, T.M., Pilipovic, S., Stankovic, B., Zorica, D.: Fractional Calculus with Application in Mechanics: Vibrations and Diffusion Processes, ISTE, London. Wiley, New York (2014)
Atanackovic, M., Pilipovic, S., Zorica, D.: Diffusion wave equation with two fractional derivatives of different order. J. Phys. A Math. Theor. 40, 5319–5333 (2007)
Atanackovic, M., Pilipovic, S., Zorica, D.: Time distributed order diffusion-wave equation part I: Volterra type equation. Proc. R. Soc. Lond. Ser. A 465, 1869–1891 (2009)
Atanackovic, T.M., Konjik, S., Pilipovic, S., Zorica, D.: Complex order fractional derivatives in viscoelasticity. Mech. Time-Depend. Mater. 20, 175–195 (2016)
Atanackovic, T.M., Janev, M., Konjik, S., Pilipovic, S.: Wave equation for generalized Zener model containing complex order fractional derivatives. Contin. Mech. Thermodyn. (2017). doi:10.1007/s00161-016-0548-4
Atanackovic, T.M., Janev, M., Pilipovic, S., Zorica, D.: Euler–Lagrange equations for Lagrangians containing complex-order fractional derivatives. J. Optim. Theory Appl. 174, 256–275 (2017)
Atanackovic, T.M., Janev, M., Konjik, S., Pilipovic, S., Zorica, D.: Expansion formula for fractional derivatives in variational problems. J. Math. Anal. Appl. 409, 911–924 (2014)
Cattaneo, C.: Del calore, sulla conduzione: Atti del Sem. Mat. Fis. Univ. di Modena 3, 83–101 (1948)
Cimmelli, V.A., Kosiński, W., Saxtion, K.: A New approach to the theory of heat conduction with finite wave speed. Le Matematiche XLVI, 95–105 (1991)
Cohen, A.M.: Numerical Methods for Laplace Transform Inversion. Springer, New York (2007)
Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A Math. Gen. 30, 7277–7289 (1997)
Crank, J.: The Mathematics of Diffusion. Clarendon Press, Oxford (1975)
Day, W.A.: The Thermodynamics of Simple Materials with Fading Memory. Springer Tracts in Natural Philosopy, vol. 22. Springer, Berlin (1972)
von Ende, S., Lion, A., Lammering, R.: On the thermodynamically consistent fractional wave equation for viscoelastic solids. Acta Mech. 221, 1–10 (2011)
Fabrizio, M., Morro, A.: Mathematical Problems in Linear Viscoelasticity. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1992)
Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)
Joseph, D.D., Preziosi, L.: Addendum to the paper, "Heat waves" [Reviews of Modern Physics, 61, 41 (1989)]. Rev. Mod. Phys. 62, 375–391 (1990)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Love, E.R.: Fractional derivatives of imaginary order. J. Lond. Math. Soc. 2(3), 241–259 (1971)
Macris, N.: Complex-parameter Kelvin model for elastic foundations. Earthq. Eng. Struct. Dyn. 23(3), 251–264 (1994)
Makris, N., Constantinou, M.: Models of viscoelasticity with complex-order derivatives. J. Eng. Mech. 119, 1453–1464 (1993)
Makris, N., Dargush, G.F., Constantinou, M.C.: Dynamic analysis of generalized viscoelastic fluids. J. Eng. Mech. 119, 1663–1679 (1993)
Fabrizio, M., Giorgi, C., Morro, A.: Modeling of heat conduction via fractional derivatives. Heat. Mass. Transf. 53, 1–13 (2017)
Morro, A.: Jump relations and discontinuity waves in conductors with memory. Math. Comput. Model. 43, 138–149 (2006)
Müller, I.: Kinetic theory and extended thermodynamics. In: Müller, I., Ruggeri, T. (eds.) Proceedings of the ISMM Symposium on Kinetic Theory and Extended Thermodynamics, Bologna, May 18–20, pp. 245–258. Pitagora Editrice, Bologna, (1987)
Müller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer Tracts in Natural Philosphy, vol. 37, second edn. Springer, New York (1998)
Ostoja-Strarzewski, M.: A derivation of the Maxwell–Cattaneo equation from the free energy and dissipation potentials. Int. J. Eng. Sci. 47, 807–810 (2009)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Povstenko, Y.: Fractional Thermoelasticity. Springer, Heidelberg (2015)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach, Amsterdam (1993)
Straughan, B.: Heat Waves. Springer, New York (2011)
Swenson, R.J.: Generalized heat conduction equation. Am. J. Phys. 46, 76–77 (1978)
Zorski, H. (ed.): Foundations of Mechanics, Studies in Applied Mechanics 28. Elsevier, Amsterdam (1992)
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Atanackovic, T.M., Pilipovic, S. On a constitutive equation of heat conduction with fractional derivatives of complex order. Acta Mech 229, 1111–1121 (2018). https://doi.org/10.1007/s00707-017-1959-4
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DOI: https://doi.org/10.1007/s00707-017-1959-4