Abstract
The classical heat conduction equation is generalized using a generalized heat conduction law. In particular, we use the space-time Cattaneo heat conduction law that contains the Caputo symmetrized fractional derivative instead of gradient \({{\partial_x}}\) and fractional time derivative instead of the first order partial time derivative \({{\partial_t}}\) . The existence of the unique solution to the initial-boundary value problem corresponding to the generalized model is established in the space of distributions. We also obtain explicit form of the solution and compare it numerically with some limiting cases.
Similar content being viewed by others
References
Atanacković, T.M., Grillo, A., Wittum, G., Zorica, D.: Fractional Jeffreys-type diffusion equation. In: Proceedings of 4th IFAC Workshop on Fractional Differentiation and Its Applications (2010)
Atanacković T.M., Pilipović S., Zorica D.: Diffusion wave equation with two fractional derivatives of different order. J. Phys. A: Math. Theor. 40, 5319–5333 (2007)
Atanacković T.M., Pilipović S., Zorica D.: Time distributed order diffusion-wave equation. I. Voltera type equation. Proc. R. Soc. A 465, 1869–1891 (2009)
Atanacković T.M., Pilipović S., Zorica D.: Time distributed order diffusion-wave equation. II. Applications of the Laplace and Fourier transformations. Proc. R. Soc. A 465, 1893–1917 (2009)
Atanacković T.M., Oparnica Lj., Pilipović S.: Semilinear ordinary differential equation coupled with distributed order fractional differential equation. Nonlinear Anal. 72, 4101–4114 (2010)
Atanackovic T.M., Stankovic B.: An expanssion formula for fractional derivatives and its application. Fractional Calc. Appl. Anal. 7(3), 365–378 (2004)
Atanacković T.M., Stanković B.: Generalized wave equation in nonlocal elasticity. Acta Mech. 208, 1–10 (2009)
Camargo R.F., Chiacchio A.O., Capelas de Oliveira E.: Differentiation to fractional orders and the fractional telegraph equation. J. Math. Phys. 49, 033505 (2008)
Camargo R.F., Charnet R., Capelas de Oliveira E.: On some fractional Green’s functions. J. Math. Phys. 50, 043514 (2009)
Cattaneo M.R.: Sur une forme de l’équation de la chaleur éliminat le paradoxe d’une propagation instantanée. Compte. Rend. 247, 431–433 (1958)
Chester M.: Second sound in solids. Phys. Rev. 131, 2013–2015 (1963)
Compte A., Metzler R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A: Math. Gen. 30, 7277–7289 (1997)
Engler, H.: On the speed of spread for fractional reaction-diffusion equations. Int. J. Differ. Equ. 2010, ID 315421 (2010)
Eringen A.C.: Nonlocal Continuum Field Theories. Springer, New York (2002)
Essex C., Schulzky C., Franz A., Hoffmann K.H.: Tsallis and Rényi entropies in fractional diffusion and entropy production. Phys. A 284, 299–308 (2000)
Hanyga A.: Multidimensional solutions of space-fractional diffusion equations. Proc. R. Soc. A 457, 2993–3005 (2001)
Hoffmann K.H., Essex C., Schulzky C.: Fractional diffusion and entropy production. J. Non-Equilib. Thermodyn. 23, 166–175 (1998)
Joseph D.D., Preziosi L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)
Konjik S., Oparnica L., Zorica D.: Waves in fractional zener type viscoelastic media. J. Math. Anal. Appl. 365, 259–268 (2010)
Kilbas A.A., Srivastava H.M., Trujillo J.J.: Theory and Applications of Fractional Differential Equations. Elsevier B.V, Amsterdam (2006)
Li X., Essex C., Davison M., Hoffmann K.H., Schulzky C.: Fractional diffusion, erreversibility and entropy. J. Non-Equilib. Thermodyn. 28, 279–291 (2003)
Mainardi F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)
Mainardi F., Pagnini G., Gorenflo R.: Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187, 295–305 (2007)
Metzler R., Klafter J.: The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–R208 (2004)
Müller I., Ruggeri T.: Rational Extended Thermodynamics (2nd edn.). Springer, Berlin (1993)
Naber, M.: Linear fractionally damped oscillator. Int. J. Differ. Equ. 2010, ID 197020 (2010)
Nakhushev, A.M.: Fractional calculus and its applications. FIZMATLIT, Moscow (2003) (in Russian)
Orsingher E., Zhao X.: The space-fractional telegraph equation and the related fractional telegraph process. Chin. Ann. Math. 24B, 45–56 (2003)
Prehl J., Essex C., Hoffmann K.H.: The superdiffusion entropy production paradox in the space-fractional case for extended entropies. Phys. A 389, 215–224 (2010)
Qi H., Jiang X.: Solutions of the space-time fractional Cattaneo diffusion equation. Phys. A 390, 1876–1883 (2011)
Ruggeri T., Muracchini A., Seccia L.: Continuum approach to phonon gas and shape changes of second sound via shock waves theory. Il Nuovo Cimento 16, 15–44 (1994)
Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon and Breach Science Publishers, Amsterdam (1993)
Silhavy M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997)
Sun HG., Chen W., Chen YQ.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A 388, 4586–4592 (2009)
Vladimirov V.S.: Equations of Mathematical Physics. Mir Publishers, Moscow (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Manuel Torrilhon.
Dedicated to Professor Ingo Müller for his 75th birthday.
Rights and permissions
About this article
Cite this article
Atanacković, T., Konjik, S., Oparnica, L. et al. The Cattaneo type space-time fractional heat conduction equation. Continuum Mech. Thermodyn. 24, 293–311 (2012). https://doi.org/10.1007/s00161-011-0199-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00161-011-0199-4