Generalized functions and additional boundary conditions in heat conduction problems for multilayered bodies

Abstract

The basic principles of a method for finding approximate analytical solutions of nonstationary heat conduction problems for multilayered structures are described. The method relies on determining a temperature perturbation front and introducing additional boundary conditions. An asymmetric unit step function is used to represent the original multilayered system as a single-layer one with piecewise homogeneous medium properties. Due to the splitting of the heat conduction process into two stages, the original partial differential equation is reduced at each stage to solving an ordinary differential equation. As a result, fairly simple (in form) analytical solutions are obtained with accuracy depending on the number of specified additional boundary conditions (on the number of approximations). It is shown that, as the number of approximations increases, same-type ordinary differential equations are obtained for the unknown time functions at the first and second stages of the process. As a result, analytical solutions can be found with a nearly prescribed degree of accuracy, including small and supersmall times.