New operational relations for mathematical models of local nonequilibrium heat transfer

Objectives. Recently, interest in studying local nonequilibrium processes has increased in the context of the development of laser technologies, the possibility of reaching ultrahigh temperatures and pressures, and the need for a mathematical description of various physical processes under extreme conditions. In simulating local nonequilibrium processes, it becomes necessary to take into account the internal structure of investigation subjects, which significantly complicates the classical transport models. An important stage here is to construct mathematical models of various physical fields in which their spatiotemporal nonlocality should be taken into account. For these purposes, hyperbolic equations are used for a wide class of phenomena and, first of all, for unsteady-state heat conduction processes based on the generalized Maxwell–Cattaneo–Luikov–Vernotte phenomenology. Mathematical models in the form of boundary value problems for hyperbolic equations are called generalized boundary value problems. These problems differ significantly in solving difficulty from the classical ones based on Fourier phenomenology. The specificity of these problems is the relative simplicity of the initial mathematical models, together with the difficulty of solving them in an analytically closed form. Hence, very little success has been achieved in finding exact analytical solutions to problems of this kind. The most acceptable approach to solving them is operational calculus. However, it gives analytical solutions in the Laplace transform space as complex functional structures, the inverse transforms of which are not available in well-known reference books on operational calculus. On this path, serious computational difficulties arise. The study aimed to analyze a set of nonstandard transforms arising from the operational solution of mathematical models of local nonequilibrium heat transfer and to obtain their inverse transforms.
Methods. Methods and theorems of operational calculus, methods of contour integration of complex transforms, and the theory of special functions were used.
Results. Operational calculus was developed for mathematical models of local nonequilibrium heat transfer in terms of the theory of unsteady-state heat conduction for hyperbolic equations (wave equations). Nonstandard operational transforms, the inverse transforms of which are unavailable in the literature, were considered. It was shown that the presented transforms are common to operational solutions of a wide class of generalized boundary value problems for hyperbolic equations in the theory of heat conduction, diffusion, hydrodynamics, vibrations, propagation of electricity, thermomechanics, and other areas of science and technology. Partially bounded and finite domains were explored. Illustrative examples were given, namely, the results of numerical experimental studies of a local nonequilibrium heat transfer process that took into account the finiteness of the heat transfer rate, which had a wave character. The latter was expressed by the presence of the Heaviside step function in the analytical solution of the problem. The physical meaning of the finiteness of the heat transfer rate was substantiated. The isochron was constructed for the temperature function in a partially bounded domain. It was shown that the temperature profile has a discontinuity on the surface of the propagating wave front. This leads to the retention of heat outflow beyond the discontinuity boundary. This is a characteristic feature of the analytical solutions of the wave equations, along with the possibility to describe the analytical solution of the problem as equivalent integral relations, which noticeably simplify numerical calculations.
Conclusions. The inverse transforms of nonstandard operational (Laplace) transforms were presented, which are contained in the operational solutions of a wide class of problems of local nonequilibrium (heat, mass, momentum) transfer processes, electrical circuits, hydrodynamics, oscillation theory, thermomechanics, and others. Illustrative examples were given, and the possibility of transition from one form of an analytical solution to another equivalent form was shown. The presented analytical solutions of hyperbolic heat transfer models in canonical domains are new in classical thermal physics.

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