Originals of operating images for generalized problems of unsteady heat conductivity

Abstract

A series of operating (Laplace) non-standard images, the originals of which are absent in well-known reference books on operational calculus, are considered. By reducing one of the basic images to the Riemann-Mellin contour integral for the modified Bessel functions and analyzing the corresponding inversion formula using the approaches of the complex variable function theory, an analytical form of the original original is found, which is abrupt in nature with a break point. It is shown that analytical solutions of the corresponding mathematical models using the found originals have a wave character, which is expressed by the presence of the Heaviside step function in the solutions. The latter means that at any time there is a region of physical disturbance to the point of discontinuity and an unperturbed area after the point of discontinuity. The images studied are included in the operational solutions of mathematical models in many areas of applied mathematics. physics, thermomechanics, thermal physics, in particular in the theory of thermal shock of viscoelastic bodies, in the study of the thermal reaction of solids based on the classical Maxwell-Cattaneo-Lykov-Vernott phenomenology, taking into account the final rate of heat propagation. These models are needed to study the thermal reaction of relatively new consolidated structurally sensitive polymeric materials in structures exposed to high-intensity external influences. The analytical relations obtained for the originals and the original improper integrals resulting from them, containing combinations of Bessel functions, can be used in the general methodology of constructing and applying various mathematical models in a wide range of external influences on materials in many fields of science and technology.