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Analytical solutions for heat diffusion beyond Fourier law. (English) Zbl 1411.35251

Summary: We obtain solutions for differential equations, describing a broad range of physical problems by the operational method with recourse to inverse differential operators, integral transforms and operational exponent. Generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions. The evolutional type problems for heat transfer in various heat conduction models are studied. Exact analytical solutions for Guyer-Krumhansl hyperbolic heat equation are obtained and compared with those of Fourier and Cattaneo equations. Modelling heat pulse propagation from a laser source is performed in the framework of Fourier, Cattaneo and Guyer-Krumhansl heat transfer models. Compliance of obtained solutions with the maximum principle is studied.

MSC:

35Q79 PDEs in connection with classical thermodynamics and heat transfer
35C05 Solutions to PDEs in closed form
35K10 Second-order parabolic equations
35L10 Second-order hyperbolic equations
78A48 Composite media; random media in optics and electromagnetic theory
80A20 Heat and mass transfer, heat flow (MSC2010)
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