Heat distribution in the rod in presence of external non-stationary source

Abstract

This paper is devoted to the study of nonlinear heat distribution in a straight homogeneous rod in the presence of an external non-stationary source of heating or cooling in relation to the process of winter concreting of columns. Until now, to describe this nonlinear distribution, as a rule, the classical linear heat equation is used. However, these models do not adequately describe the real process, since the nonlinearity of the process and the presence of an external heat source are not taken into account. Using group analysis methods, we obtained a model that admits the widest group of Lie transformations compared to other basic models of the general model. For the differential equation defining this model, we have obtained all separable solutions and some invariant solutions. The set of these solutions depends on empirically determined parameters: one arbitrary smooth function and ten arbitrary constants. Each solution determines a new exact model for winter concreting of columns. For each solution, at certain values of the parameters on which this solution depends, the temperature distribution in the rod were obtained. The significance of the obtained solutions is as follows: 1) these solutions describe specific physical processes and can be used in practice, 2) these solutions can be used as test solutions in numerical calculations.