Invariant nonlinear heat distribution in a rod in the presence of an external nonstationary source of heating or cooling

This paper is devoted to the study of nonlinear heat distribution in a homogeneous rod in the presence of an external nonstationary source of heating or cooling. We used a parametric function characterizing temperature. Since temperature is a monotonically increasing function of this temperature parameter, the nature of the behavior of temperature coincides with the nature of the behavior of this parameter. An equation for the temperature parameter is obtained. This equation defines the main model for the study performed. Using symmetry analysis methods, all its basic models of this model were found, which have different symmetry properties. Further research is devoted to the model that allows the widest group of Lie transformations compared to other basic models. For the differential equation defining this model, we have obtained all separable solutions and all invariant solutions. Some solutions are found explicitly. The set of these explicit solutions depends on empirically determined parameters: any one smooth function and ten arbitrary constants. For other solutions, we studied some physically significant boundary value problems. Boundary value problems for some specific values of the parameters included in them are solved numerically.