Home – Home – Jornals – Journal of Applied Mechanics and Technical Physics 2025 number 2

An unsteady problem of heat conduction for a rod is considered. The classical heat conduction equation based on the assumption of temperature differentiability with respect to time and coordinate is derived. A solution of a model problem with boundary conditions of the second kind is obtained, which determines the temperature distribution in the thermally insulated rod over its length and in time. For the classical formulation of the problem, the temperature change rate at the initial time is found to be singular, and the condition of temperature differentiability with respect to time is not satisfied. A modified form of the heat conduction equation is proposed, which is based on nonlocal determination of temperature as a time-dependent function. In contrast to the traditional definition of temperature, this function is not the temperature value at a fixed time; instead, it is the mean value on a finite time interval called the nonlocal temperature. As a result of using such an approach, the heat conduction equation retains the classical form, but contains the nonlocal temperature rather than the traditionally used temperature. Traditionally, the temperature is determined by means of solving the Helmholtz equation including an unknown time interval over which the temperature is averaged and which is determined experimentally. The classical and nonlocal solutions are compared with experimental data. The nonclassical Maxwell--Cattaneo equation of heat conduction is discussed, which implies a finite rate of temperature propagation in time.