Time-Fractional Cattaneo-Type Thermoelastic Interior-Boundary Value Problem Within A Rigid Ball

Time-Fractional Cattaneo-Type Thermoelastic Interior-Boundary Value Problem Within A Rigid Ball

The paper discusses the solution of an interior-boundary value problem of one-dimensional time-fractional Cattaneo-type heat conduction and its stress fields for a rigid ball. The interior value problem describes the dependence of the boundary conditions within the ball's inner plane at any instant with a prescribed temperature state, in contrast to the exterior value problem, which relates the known surface temperature to boundary conditions. A single-phase-lag equation with Caputo fractional derivatives is proposed to model the heat equation in a medium subjected to time-dependent physical boundary conditions. The application of the finite spherical Hankel and Laplace transform technique to heat conduction is discussed. The influence of the fractional-order parameter and the relaxation time is examined on the temperature fields and their related stresses. The findings show that the slower the thermal wave, the bigger the fractional-order setting, and the higher the period of relaxation, the slower the heat flux propagates.

___

  • G. Stolz, “Numerical solutions to an inverse problem of heat conduction for simple shapes,” J. Heat Transfer, 82, 20-25, 1960. DOI: 10.1115/1.3679871.
  • D. Necsulescu (2009) Advanced Mechatronics: Monitoring and Control of Spatially Distributed Systems, Singapore: World Scientific Company.
  • K. Woodbury (2003) Inverse Engineering Handbook, Boca Raton: CRC Press.
  • M. Ozisik and H. Orlande (2000) Inverse Heat Transfer: Fundamentals and Applications, New York ; Taylor & Francis.
  • J. Beck, B. Blackwell and JR Charles (1985) Inverse Heat Conduction: Ill-posed Problems. New York; John Willey & Sons.
  • L. Torsten, A. Mhamdi, and W. Marquardt, “Design, formulation, and solution of multidimensional inverse heat conduction problems,” Numer. Heat Tr-B Fund., 47, 111 - 133, 2005. DOI: 10.1080/10407790590883351.
  • X. Lu and P. Tervola, “Transient heat conduction in the composite slab-analytical method,” J Phys Math Gen, 38, 81-96, 2005. DOI: 10.1088/0305-4470/38/1/005.
  • K. W. Khobragade, V. Varghese and N. W. Khobragade, “An inverse transient thermoelastic problem of a thin annular disc,” Appl. Math. E-Notes, 6, 17-25, 2006.
  • P. L. Woodfield, M. Monde, and Y. Mitsutake, “Improved analytical solution for inverse heat conduction problems on thermally thick and semi-infinite solids,” Int. J. Heat Mass Transfer, 49, 2864-2876, 2006. DOI: 10.1016/j.ijheatmasstransfer.2006.01.050.
  • R. Pourgholi, and M. Rostamian, “A numerical technique for solving IHCPs using Tikhonov Regularization Method,” Appl. Math. Model, 34, 2102-2110, 2010. DOI: 10.1016/j.apm.2009.10.022.
  • S. Danaila, and A. Chira, “Mathematical and numerical modeling of inverse heat conduction problem,” INCAS BULLETIN, 6, 23–39, 2014. DOI: 10.13111/2066-8201.2014.6.4.3.
  • M. Ivanchov and N. Kinash, “Inverse problem for the heat-conduction equation in a rectangular domain,” Ukr. Math. J., 69, 2018.DOI: 10.1007/s11253-018-1476-1.
  • H. Chen, I. Jay, J. Frankel, and M. Keyhani, “Nonlinear inverse heat conduction problem of surface temperature estimation by calibration integral equation method,” Numer. Heat Tr-B Fund., 73, 263–291, 2018. DOI: 10.1080/10407790.2018.1464316.
  • C. Chang, C. Liu, and C. Wang, “Review of computational schemes in inverse heat conduction problems,” Smart Sci., 6, 94–103, 2018. DOI: 10.1080/23080477.2017.1408987.
  • S. Kukla and U. Siedlecka, “Laplace transform solution of the problem of time-fractional heat conduction in a two-Layer slab,” J. Appl. Comput. Mech., 14, 105-113, 2015.DOI: 10.17512/jamcm.2015.4.10.
  • E. Hoashi, T. Yokomine, A. Shimizu, and T. Kunugi, “Numerical analysis of wave-type heat transfer propagating in a thin foil irradiated by short-pulsed laser,” Int. J. Heat Mass Transf., 46, 4083–4095, 2003. DOI: 10.1016/S0017-9310(03)00225-4.
  • X. Ai and B. Q. Li, “Numerical simulation of thermal wave propagation during laser processing of thin films,” J. Electron. Mater., 34, 583–591, 2005. DOI: 10.1007/s11664-005-0069-6.
  • T. T. Lam and E. Fong, “Application of solution structure theorem to non-Fourier heat conduction problems: Analytical approach,” Int. J. Heat Mass Transf., 54, 4796–4806, 2011. DOI: 10.1016/j.ijheatmasstransfer.2011.06.028.
  • T. T. Lam, “A unified solution of several heat conduction models,” Int. J. Heat Mass Transf., 56, 653–666, 2013. DOI: 10.1016/j.ijheatmasstransfer.2012.08.055.
  • C. Cattaneo, “Sur uneForme de I’equation de la Chaleur Eliminant le Paradoxed’une Propagation Instantanee’,” ComptesRendus de l’Académie des Sciences, 247, 431-433, 1958.
  • P. Vernotte, “Les paradoxes de la théorie continue de l’équation de la chaleur,” C. R. Acad. Sci. Paris, 246, 3154-3155, 1958.
  • A. Compte and R. Metzier, “The generalized Cattaneo equation for the description of anomalous transport processes,” J. Phys. A: Math. Gen., 30, 7277-7289, 1997.
  • Y. Z. Povstenko, “Fractional Cattaneo-type equations and generalized thermoela-sticity,” J. Therm. Stresses, 34, 97-114, 2011. DOI: 10.1080/01495739.2010.511931
  • T. N. Mishra and K. N. Rai, “Numerical solution of FSPL heat conduction equation for analysis of thermal propagation,” Appl. Math. Comput., 273, 1006–1017, 2016. DOI: 10.1016/j.amc.2015.10.082.
  • H.-T. Qi, H.-Y. Xu, and X.-W. Guo, “The Cattaneo-type time fractional heat conduction equation for laser heating,” Comput. Math. Appl., 66, 824–831, 2013. DOI: 10.1016/j.camwa.2012.11.021.
  • H. T. Qi, H. Y. Xu and X. W. Guo, “The Cattaneo-type time fractional heat conduction equation for laser heating,” Comput. Math. Appl., 66, 824–831, 2013. DOI: 10.1016/j.camwa.2012.11.021
  • H. T. Qi, and X.W. Guo, “Transient fractional heat conduction with generalized Cattaneo model,” Int. J. Heat Mass Transfer, 76, 535–539, 2014. DOI: 10.1016/j.ijheatmasstransfer.2013.12.086.
  • H. Y. Xu, H. T. Qi, and X. Y. Jiang, “Fractional Cattaneo heat equation in a semi-infinite medium,” Chin. Phys. B, 22, 014401, 2013. DOI: 10.1088/1674-1056/22/1/014401
  • G. Y. Xu, J. B. Wang, and Z. Han, “Study on the transient temperature field based on the fractional heat conduction equation for laser heating,” Appl. Math. Mech., 36, 844–849, 2015.
  • H. R. Ghazizadeh, M. Maerefat, and A. Azimi, “Explicit and implicit finite difference schemes for fractional Cattaneo equation,” J. Comput. Phys., 229, 7042–7057, 2010. DOI: 10.1016/j.jcp.2010.05.039.
  • M. N. Özisik (1993) Heat Conduction, John Wiley & Sons, New York.
  • D. Y. Tzou, “Thermal shock phenomena under high rate response in solids,” Ann. Rev. Heat Transf., 4, 111–185, 1992. DOI: 10.1615/AnnualRevHeatTransfer.v4.50
  • M. E. Gurtin and A. C. Pipkin, “A general theory of heat conduction with finite wave speeds,” Arch. Rational Mech. Anal., 31, 113–126, 1968. DOI: 10.1007/BF00281373
  • A. Compte, A. Metzler, and J. Camacho, “Biased continuous time random walks between parallel plates,” Phys. Rev. E, 56, No. 2, 1445-1454, 1997.
  • M. N. Özisik, D. Y. Tzou, “On the wave theory in heat conduction,” J. Heat Transf., 116, 526–535, 1994.DOI: 10.1115/1.2910903
  • S. Kaliski, “Wave equations of thermoelasticity,” Bull. Acad. Polon. Sci. Ser. Sci. Techn., 13, 253-260, 1965.
  • H. W. Lord and Y. Shulman, “A generalized dynamical theory of thermoelasticity,” J. Mech. Phys. Solids, 15, 299-309, 1967.
  • I. Podlubny (1999) Fractional Differential Equations, Academic Press, New York.
  • Y. Povstenko, “Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition,” Eur. Phys. J. Spec. Top., 222, 1767–1777, 2013. DOI: 10.1140/epjst/e2013-01962-4.
  • Y. Z. Povstenko, “Axisymmetric Solutions to Time-Fractional Heat Conduction Equation in a Half-Space under Robin Boundary Conditions,” Int. J. Differ. Equ., 2012, 1–13, 2012. DOI: 10.1155/2012/154085.
  • Y. Povstenko, “Fundamental solutions to the fractional heat conduction equation in a ball under Robin boundary condition,” Open Math., 12, 2014. DOI: 10.2478/s11533-013-0368-8.
  • H. Beyer and S. Kempfle, “Definition of physically consistent damping laws with fractional derivatives,” ZAMM - J. Appl. Math. Mech., 75, 623–635, 1995. DOI: 10.1002/zamm.19950750820.
  • N. Noda, R. B. Hetnarski, and Y. Tanigawa (2003) Thermal stresses, 2nd Ed., Taylor and Francis, New York, 2003.
  • B.E. Ghonge and K.P. Ghadle, “An inverse transient thermoelastic problem of solid sphere,” Bulletin of Pure and Applied Sciences, 29, 1-9, 2010.
  • I. N. Snedden (1972) The Use of Integral Transforms, McGraw-Hill Book Co., New York.
  • I. H. Chen, “Modified Fourier‐Bessel series and finite spherical Hankel transform,” Int. J. Math. Educ. Sci. Technol., 13, No. 3, 281-283, 1982. DOI: 10.1080/0020739820130307
  • S. N. Li, B.Y. Cao, “Fractional Boltzmann transport equation for anomalous heat transport and divergent thermal conductivity,” Int. J. Heat Mass Transf., 137, 84-89, 2019.
  • S. N. Li, B.Y. Cao, “Fractional-order heat conduction models from generalized Boltzmann transport equation,” Philos. Trans. R. Soc. A, 378, 20190280, 2020.