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Exact and analytic-numerical solutions of bidimensional lagging models of heat conduction. (English) Zbl 1235.80046

Summary: Lagging models of heat conduction, such as the Dual-Phase-Lag or the Single-Phase-Lag models, lead to heat conduction equations in the form of partial differential equations with delays or to partial differential equations of hyperbolic type, and have been considered to model microscale heat transfer in engineering problems or bio-heat transfer in medical treatments.

MSC:

80M25 Other numerical methods (thermodynamics) (MSC2010)
80A20 Heat and mass transfer, heat flow (MSC2010)
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35C10 Series solutions to PDEs
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[1] Joseph, D.D.; Preziosi, L., Heat waves, Rev. modern phys., 61, 41-73, (1989) · Zbl 1129.80300
[2] Tzou, D.Y., Macro- to microscale heat transfer: the lagging behavior, (1996), Taylor & Francis Washington
[3] Tzou, D.Y., On the thermal shock wave induced by a moving heat source, J. heat transfer, 111, 232-238, (1989)
[4] Tzou, D.Y., The generalized lagging response in small-scale and high-rate heating, Int. J. heat mass transfer, 38, 3231-3240, (1995)
[5] Tzou, D.Y., Experimental support for the lagging behavior in heat propagation, AIAA J. termophys. heat transfer, 9, 686-693, (1995)
[6] Qiu, T.Q.; Tien, C.L., Short-pulse laser heating on metals, Int. J. heat mass transfer, 35, 719-726, (1992)
[7] Qiu, T.Q.; Tien, C.L., Heat transfer mechanisms during short-pulse laser heating of metals, ASME J. heat transfer, 115, 835-841, (1993)
[8] Wang, L.; Wei, X., Heat conduction in nanofluids, Chaos solitons fractals, 39, 2211-2215, (2009)
[9] Vadasz, J.J.; Govender, S., Thermal wave effects on heat transfer enhancement in nanofluids suspensions, Int. J. thermal sci., 49, 235-242, (2010)
[10] Xu, F.; Seffen, K.A.; Liu, T.J., Non-Fourier analysis of skin biothermomechanics, Int. J. heat mass transfer, 51, 2237-2259, (2008) · Zbl 1144.80358
[11] Zhou, J.; Zhang, Y.; Chen, J.K., An axisymmetric dual-phase-lag bioheat model for laser heating of living tissues, Int. J. thermal sci., 48, 1477-1485, (2009)
[12] Zhou, J.; Chen, J.K.; Zhang, Y., Dual-phase-lag effects on thermal damage to biological tissues caused by laser irradiations, Comput. biol. med., 39, 286-293, (2009)
[13] Cattaneo, C., Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée, C. R. acad. sci., 247, 431-433, (1958) · Zbl 1339.35135
[14] Vernotte, P., LES paradoxes de la théorie continue de l’équation de la chaleur, C. R. acad. sci., 246, 3154-3155, (1958) · Zbl 1341.35086
[15] Vernotte, P., Some possible complications in the phenomena of thermal conduction, C. R. acad. sci., 252, 2190-2191, (1961)
[16] Tzou, D.Y., A unified approach for heat conduction from macro to micro-scales, ASME J. heat transfer, 117, 8-16, (1995)
[17] Quintanilla, R.; Racke, R., A note on stability in dual-phase-lag heat conduction, Int. J. heat mass transfer, 49, 1209-1213, (2006) · Zbl 1189.80025
[18] Kulish, V.V.; Novozhilov, V.B., An integral equation for the dual-lag model of heat transfer, ASME J. heat transfer, 126, 805-808, (2004)
[19] Xu, M.; Wang, L., Dual-phase-lagging heat conduction based on Boltzmann transport equation, Int. J. heat mass transfer, 48, 5616-5624, (2005) · Zbl 1188.76242
[20] Martín, J.A.; Rodríguez, F.; Company, R., Analytic solution of mixed problems for the generalized diffusion equation with delay, Math. comput. modelling, 40, 361-369, (2004) · Zbl 1062.35157
[21] Reyes, E.; Rodríguez, F.; Martín, J.A., Analytic-numerical solutions of diffusion mathematical models with delays, Comput. math. appl., 56, 743-753, (2008) · Zbl 1155.65387
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