Nonlocal Solution to the Heat Conducton Problem for a Rod

An unsteady heat conduction problem for a rod is considered. A classical heat conduction equation is obtained on the basis of the assumption of temperature differentiability with respect to time and coordinate. A solution is constructed for a model problem with boundary conditions of the second kind, which determines temperature distribution in a heat-insulated rod along its length and time. It is revealed that, for the classical formulation of the problem, the temperature variation rate at the initial moment is singular and the condition of temperature differentiability with respect to time is not satisfied. A modified heat conduction equation, which is based on a nonlocal definition of temperature as a time-dependent function, is proposed. Unlike the traditional definition of temperature, this function is not the temperature value at a fixed moment, but rather is an average value over a finite time interval, known as a nonlocal temperature. With the application of this approach, the heat conduction equation retains the classical form, but contains the nonlocal temperature rather than the traditionally used. As a rule, temperature is determined by solving the Helmholtz equation, which includes an unknown time interval over which the temperature is averaged and which is determined experimentally. The classical and the nonlocal solution are compared to experimental data. This paper also touches upon the Maxwell–Cattaneo nonclassical law of heat conduction, which assumes a finite temperature propagation velocity over time.