Far Eastern Mathematical Journal

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The stability of steady-state solutions of the diffusion complex heat transfer model


Grenkin G. V., Chebotarev A.Yu.

2014, issue 1, Ñ. 18-32


Abstract
The nonstationary model of radiative-convective-conductive heat transfer in a three-dimensional domain within the diffusion P1 approximation of radiative transfer is considered. The sufficient conditions of asymptotic stability of steady states are established.

Keywords:
radiative heat transfer equations, diffusion approximation, nonlocal solvability, asymptotic stability

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References

[1] D. A. Boas, Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications, A Ph.D. Dissertation in Physics, University of Pennsylvania, 1996.
[2] M. F. Modest, Radiative Heat Transfer, Academic Press, 2003.
[3] S. E. Siewert, “An improved iterative method for solving a class of coupled conductiveradiative heat-transfer problems”, J. Quant. Spectrosc. Radiat. Transfer., 54:4 (1995), 599–605.
[4] J. M. Banoczi, C. T. Kelley, “A fast multilevel algorithm for the solution of nonlinear systems of conductive-radiative heat transfer equations”, SIAM J. Sci. Comput., 19:1 (1998), 266–279.
[5] A. Klar, N. Siedow, “Boundary layers and domain decomposition for radiative heat transfer and diffusion equations: applications to glass manufacturing process”, Eur. J. Appl. Math., 9:4 (1998), 351–372.
[6] G. Th?omes, R. Pinnau, M. Sea?id, T. G?otz, A. Klar, “Numerical methods and optimal control for glass cooling processes”, Trans. Theory Stat. Phys., 31:4-6 (2002), 513–529.
[7] R. Pinnau, M. Seaid, “Simplified PN Models and Natural Convection–Radiation”, Math. in Industry, V. 12 (2008), 397–401.
[8] A. E. Kovtanyuk, N. D. Botkin, K.-H. Hoffmann, “Numerical simulations of a coupled conductive-radiative heat transfer model using a modified Monte Carlo method”, Int. J. Heat and Mass Transfer., 55 (2012), 649–654.
[9] A. E. Kovtaniuk, “Algoritmy parallel'nykh vychislenii dlia zadach radiatsionno-konduktivnogo teploobmena”, Komp'iuternye issledovaniia i modelirovanie, 4:3 (2012), 543–552.
[10] A. E. Kovtanyuk, A.Yu. Chebotarev, “An iterative method for solving a complex heat transfer problem”, Appl. Math. Comput., 219 (2013), 956–9362.
[11] A. A. Amosov, “Global'naia razreshimost' odnoi nelineinoi nestatsionarnoi zadachi s nelokal'nym kraevym usloviem tipa teploobmena izlucheniem”, Differentsial'nye uravneniia, 41:1 (2005), 93–104.
[12] R. Pinnau, “Analysis of Optimal Boundary Control for Radiative Heat Transfer Modelled by the SP1-System”, Comm. Math. Sci., 5:4 (2007), 951–969.
[13] P.-E. Druet, “Existence of weak solutions to the time-dependent MHD-equations coupled to heat transfer with nonlocal radiation boundary conditions”, Nonlinear Anal. Real World Appl., 10:5 (2009), 2914–2936.
[14] B. Ducomet, S. Necasova, “Global Weak Solutions to the 1D Compressible Navier-Stokes Equations with Radiation”, Commun. Math. Anal., 8:3 (2010), 23–65.
[15] O. Tse, R. Pinnau, N. Siedow, “Identification of temperature dependent parameters in laser–interstitial thermo therapy”, Math. Models Methods Appl. Sci., 22:9 (2012), 1–29.
[16] C. T. Kelley, “Existence and uniqueness of solutions of nonlinear systems of conductiveradiative heat transfer equations”, Transport Theory Statist. Phys., 25:2 (1996), 249–260.
[17] A. E. Kovtanyuk, A.Yu. Chebotarev, N. D. Botkin, K.-H. Hoffmann, “The unique solvability of a complex 3D heat transfer problem”, J. Math. Anal. Appl., 409:2 (2014), 808–815.
[18] A. E. Kovtanyuk, A.Yu. Chebotarev, N. D. Botkin, K.-H. Hoffmann, “Theoretical analysis of an optimal control problem of conductive-convective-radiative heat transfer”, J. Math. Anal. Appl., 412:1 (2014), 520–528.
[19] A. E. Kovtaniuk, A.Iu. Chebotarev, “Statsionarnaia zadacha slozhnogo teploobmena”, Zh. vychisl. matem. fiz., 54:4 (2014), 191-199.
[20] A. A. Amosov, “O razreshimosti odnoi zadachi teploobmena izlucheniem”, Dokl. AN SSSR, 245:6 (1979), 1341–1344.
[21] M. T. Laitinen, T. Tiihonen, “Heat transfer in conducting, radiating and semitransparent materials”, Math. Meth. Appl. Sci., 21 (1998), 375–392.
[22] M. Laitinen, “Asymptotic analysis of conductive-radiative heat transfer”, Asymptotic Analysis, 29:3-4 (2002), 323–342.
[23] A. A. Amosov, “Stationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency”, Journal of Mathematical Sciences, 164:3 (2010), 309–344.
[24] A. A. Amosov, “Nonstationary nonlinear nonlocal problem of radiative-conductive heat transfer in a system of opaque bodies with properties depending on the radiation frequency”, Journal of Mathematical Sciences, 165:1 (2010), 1–41.
[25] Partial Differential Equations — MATLAB & Simulink. http://www.mathworks.com/help/matlab/math/partial-differential-equations.html.

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