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Time-fractional heat conduction in a two-layer composite slab

  • Yuriy Povstenko EMAIL logo
Published/Copyright: August 27, 2016
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 4

Abstract

The heat conduction equation is considered in a composite body consisting of two regions: 0 < x < L and − L < x < 0. Heat conduction in one region is described by the equation with the Caputo fractional derivative of order α, whereas in another region by the equation with the Caputo fractional derivative of order β. The integral transforms technique is used. The approximate solution valid for small values of time is analyzed in detail.

MSC 2010: Primary 26A33; Secondary 35K05, 45K05
Key Words and Phrases: fractional calculus; Mittag-Leffler type functions; fractional partial differential equations; Laplace transform

References

[1] A. Erdélyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).Search in Google Scholar

[2] V. Gafiychuk, B. Datsko, Mathematical modeling of different types of instabilities in time fractional reaction-diffusion systems. Comput. Math. Appl. 59, No 3 (2010), 1101–1107.10.1016/j.camwa.2009.05.013Search in Google Scholar

[3] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, New York (2014).10.1007/978-3-662-43930-2Search in Google Scholar

[4] R. Gorenflo, J. Loutchko, Yu. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal. 5, No 4 (2002), 491–518.Search in Google Scholar

[5] R. Gorenflo, Yu. Luchko, F. Mainardi, Analytical properties and applications of the Wright function. Fract. Calc. Appl. Anal. 2, No 4 (1999), 383–414.Search in Google Scholar

[6] R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien (1997), 223–276.10.1007/978-3-7091-2664-6_5Search in Google Scholar

[7] J. Ignaczak, M. Ostoja-Starzewski, Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2009).10.1093/acprof:oso/9780199541645.001.0001Search in Google Scholar

[8] D.D. Joseph, L. Preziosi, Heat waves. Rev. Mod. Phys. 61, No 1 (1989), 41–73.10.1103/RevModPhys.61.41Search in Google Scholar

[9] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Search in Google Scholar

[10] S. Kukla, U. Siedlecka, Laplace transform solution of the problem of time-fractional heat conduction in a two-layered slab. J. Appl. Math. Comput. Mech. 14, No 4 (2015), 105–113.10.17512/jamcm.2015.4.10Search in Google Scholar

[11] Yu. Luchko, Algorithms for evaluation of the Wright function for the real arguments’ values. Fract. Calc. Appl. Anal. 11, No 1 (2008), 57–75; at http://www.math.bas.bg/~fcaa.Search in Google Scholar

[12] A.V. Luikov, Analytical Heat Diffusion Theory. Academic Press, New York (1968).Search in Google Scholar

[13] R.L. Magin, Fractional Calculus in Bioengineering. Begell House Publishers, Connecticut (2006).Search in Google Scholar

[14] F. Mainardi, The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9, No 6 (1996), 23–28.10.1016/0893-9659(96)00089-4Search in Google Scholar

[15] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos, Solitons Fractals7, No 9 (1996), 1461–1477.10.1016/0960-0779(95)00125-5Search in Google Scholar

[16] F. Mainardi, Fractional calculus: Some basic problems in continuum and statistical mechanics. In: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien (1997), 291–348.10.1007/978-3-7091-2664-6_7Search in Google Scholar

[17] R. Metzler, J. Klafter, 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, No 31 (2004), R161–R208.10.1088/0305-4470/37/31/R01Search in Google Scholar

[18] M.N. Özişik, Heat Conduction. John Wiley, New York (1980).Search in Google Scholar

[19] N. Petrov, G. Brankov, Modern Problems of Thermodynemics. Publishing House of Bulgarian Academy of Sciences, Sofia (1982), In Bulgarian.Search in Google Scholar

[20] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Search in Google Scholar

[21] Y. Povstenko, Fractional heat conduction equation and associated thermal stresses. J. Thermal Stresses28, No 1 (2005), 83–102.10.1080/014957390523741Search in Google Scholar

[22] Y. Povstenko, Thermoelasticity which uses fractional heat conduction equation. J. Math. Sci. 162, No 2 (2009), 296–305 (see also Math. Meth. Phys.-Mech. Fields51, No 2 (2008), 239–246).10.1007/s10958-009-9636-3Search in Google Scholar

[23] Y. Povstenko, Theory of thermoelasticity based on the space-time-fractional heat conduction equation. Phys. Scr. T. 136 (2009), 014017–1–6.10.1088/0031-8949/2009/T136/014017Search in Google Scholar

[24] Y. Povstenko, Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder. Fract. Calc. Appl. Anal. 14, No 3 (2011), 418–435; 10.2478/s13540-011-0026-4; http://www.degruyter.com/view/j/fca.2011.14.issue-3/issue-files/fca.2011.14.issue-3.xml.Search in Google Scholar

[25] Y. Povstenko, Different kinds of boundary problems for fractional heat conduction equation. In: I. Petráš, I. Podlubny, J. Kostúr, J. Kačur and A. MojžiŠová (Eds.), Proc. 13th Int. Carpathian Control Conf., Hight Tatras, Slovakia, 28-31 May 2012, Institute of Electrical and Electronics Engineers, KoŠice (2012), 588–591.10.1109/CarpathianCC.2012.6228713Search in Google Scholar

[26] Y. Povstenko, Fractional heat conduction in infinite one-dimensional composite medium. J. Thermal Stresses36, No 4 (2013), 351–363.10.1080/01495739.2013.770693Search in Google Scholar

[27] Y. Povstenko, Fractional heat conduction in an infinite medium with a spherical inclusion. Entropy15, No 10 (2013), 4122–4133.10.3390/e15104122Search in Google Scholar

[28] Y. Povstenko, Fundamental solutions to time-fractional heat conduction equations in two joint half-lines. Cent. Eur. J. Phys. 11, No 19 (2013), 1284–1294.10.2478/s11534-013-0272-7Search in Google Scholar

[29] Y. Povstenko, Fractional heat conduction in a semi-infinite composite body. Comm. Appl. Industr. Math. 6 (2014), 1–13; 10.1685/journal.caim.482.Search in Google Scholar

[30] Y. Povstenko, Fractional Thermoelasticity. Springer, New York (2015).10.1007/978-3-319-15335-3Search in Google Scholar

[31] Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. Birkhäuser, New York (2015).10.1007/978-3-319-17954-4Search in Google Scholar

[32] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, Vol. 1: Elementary Functions. Gordon and Breach Science Publishers, Amsterdam (1986).Search in Google Scholar

[33] I.N. Sneddon, The Use of Integral Transforms. McGraw-Hill, New York (1972).Search in Google Scholar

[34] B. Stanković, On the function of E. M. Wright. Publ. Inst. Math. 10, No 24 (1970), 113–124.Search in Google Scholar

[35] K.K Tamma, X. Zhou, Macroscale thermal transport and thermomechanical interaction: some noteworthy perspectives. J. Thermal Stresses21, No 3-4 (1998), 405–449.10.1080/01495739808956154Search in Google Scholar

[36] V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers. Springer, Berlin (2013).10.1007/978-3-642-33911-0Search in Google Scholar

Received: 2015-12-29
Published Online: 2016-8-27
Published in Print: 2016-8-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–4–2016)
  4. Survey Paper
  5. Responses comparison of the two discrete-time linear fractional state-space models
  6. Survey Paper
  7. A survey on impulsive fractional differential equations
  8. Research Paper
  9. Generalization of the fractional poisson distribution
  10. Research Paper
  11. Diffusivity identification in a nonlinear time-fractional diffusion equation
  12. Research Paper
  13. An extension problem for the fractional derivative defined by Marchaud
  14. Research Paper
  15. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
  16. Research Paper
  17. The Neumann problem for the generalized Bagley-Torvik fractional differential equation
  18. Research Paper
  19. On the fractional probabilistic Taylor's and mean value theorems
  20. Research Paper
  21. Time-fractional heat conduction in a two-layer composite slab
  22. Research Paper
  23. Weighted adams type theorem for the riesz fractional integral in generalized morrey space
  24. Research Paper
  25. Fractional schrödinger equation with zero and linear potentials
  26. Research Paper
  27. Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions
  28. Research Paper
  29. Weighted bounded solutions for a class of nonlinear fractional equations
  30. Research Paper
  31. Existence and global asymptotic behavior of positive solutions for superlinear fractional dirichlet problems on the half-line
  32. Short Paper
  33. Functional delay fractional equations
https://doi.org/10.1515/fca-2016-0051
Keywords for this article
fractional calculus; Mittag-Leffler type functions; fractional partial differential equations; Laplace transform

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA Related News, Events and Books (FCAA–Volume 19–4–2016)
  4. Survey Paper
  5. Responses comparison of the two discrete-time linear fractional state-space models
  6. Survey Paper
  7. A survey on impulsive fractional differential equations
  8. Research Paper
  9. Generalization of the fractional poisson distribution
  10. Research Paper
  11. Diffusivity identification in a nonlinear time-fractional diffusion equation
  12. Research Paper
  13. An extension problem for the fractional derivative defined by Marchaud
  14. Research Paper
  15. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem
  16. Research Paper
  17. The Neumann problem for the generalized Bagley-Torvik fractional differential equation
  18. Research Paper
  19. On the fractional probabilistic Taylor's and mean value theorems
  20. Research Paper
  21. Time-fractional heat conduction in a two-layer composite slab
  22. Research Paper
  23. Weighted adams type theorem for the riesz fractional integral in generalized morrey space
  24. Research Paper
  25. Fractional schrödinger equation with zero and linear potentials
  26. Research Paper
  27. Positive solutions of higher-order nonlinear multi-point fractional equations with integral boundary conditions
  28. Research Paper
  29. Weighted bounded solutions for a class of nonlinear fractional equations
  30. Research Paper
  31. Existence and global asymptotic behavior of positive solutions for superlinear fractional dirichlet problems on the half-line
  32. Short Paper
  33. Functional delay fractional equations
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