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Axioms 2018, 7(3), 48; https://doi.org/10.3390/axioms7030048

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

Faculty of Physics, M.V. Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia
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Received: 10 April 2018 / Revised: 3 July 2018 / Accepted: 13 July 2018 / Published: 18 July 2018
(This article belongs to the Special Issue Mathematical Analysis and Applications)
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Abstract

One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed. View Full-Text
Keywords: exponential operator; differential operator; Guyer–Krumhansl equation; moving media; non–Fourier; heat conduction; Knudsen number exponential operator; differential operator; Guyer–Krumhansl equation; moving media; non–Fourier; heat conduction; Knudsen number
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Zhukovsky, K.; Oskolkov, D.; Gubina, N. Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative. Axioms 2018, 7, 48.

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