Parabolic Profile in Heat-Conduction Problems. 3. Semi-Bounded Space with a Definite External Heat Flow

Abstract

Some approaches to the optimization of the exponent n of the parabolic temperature profile for a semi-bounded space with the second-kind boundary condition, implying the prescription of an external heat flow, are considered. An optimization scheme involving the minimization of a new error norm estimating the absolute value of a temperature deviation, more efficient than the scheme of T. Myers with minimization of the Langford norm, as well as new hybrid integral methods are proposed. With the use of these methods, parabolic solutions have been obtained on the basis of the temperature momentum integral in different modifications and the heat-flow momentum integral, and these solutions define the surface temperature with a very high accuracy. As an illustrative example, the problem on the heat conduction of a semi-bounded space with a definite pulsating heat flow at its surface, whose time change is defined by a quadratic parabola, was considered. The approximation errors of the parabolic solutions obtained using the hybrid integral methods proposed comprise hundredths and thousandths of a percent, and they are smaller by one to two orders of magnitude than the errors of the analogous solutions obtained by the known heat balance integral method and refined integral method, involving the optimization of the exponent of a parabolic temperature profile on the basis of minimization of the Langford norm, and by the combined integral method.