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.
Highlights
Recent progress in technology and science has driven interest to studies of heat conduction beyond common Fourier law [1]: ∂t T = k∂2x T, where k is the thermal diffusivity, T is the temperature; Fourier law describes heat conduction in homogeneous matter at normal conditions well
That for some values of the coefficients α and δ the quantities U and u can assume negative values. Albeit it is not very obvious in the form of the harmonic solution (27) to Guyer and Krumhansl (GK)-type equation with substantial derivative (26) and the solution (31) to TE with substantial derivative (28), these solutions remains real at any moment of time for a real initial function, because of the complex exponential is compensated by proper complex parts in the coefficients C1,2 and B1,2
In order to obtain the exact solution to this problem we reduced GK-type Equation (9) in the harmonic ansatz to
Summary
Recent progress in technology and science has driven interest to studies of heat conduction beyond common Fourier law [1]: ∂t T = k∂2x T, where k is the thermal diffusivity, T is the temperature; Fourier law describes heat conduction in homogeneous matter at normal conditions well. More complicated system of equations was proposed for the ballistic heat transport in [30], while heat propagation in highly inhomogeneous materials was reported to be close to GK law [9,11]. We will analyze the solutions for the heat transport, described by the system of inhomogeneous partial derivative equation, and use the Knudsen number to account for the ballistic conditions in 1- and 2-dimensional structures [70]
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