A new approach to approximate solution of the Stefan problem with a convective boundary condition (Communicated by Corresponding Member Nikolai V. Pavlyukevich)

Abstract

Two new variants of approximate analytical solution of the one-phase Stefan problem with a convective boundary condition at a fixed boundary are proposed. These approaches are based on the use of new integral relations forming infinite sequences. It is shown that the most exact variant of solving the Stefan problem with a convective boundary condition is to refuse from the classical Stefan condition at the free boundary and to replace it with its integral relation. By the example of solving the test Stefan problem with a Robin boundary condition, having an exact analytical solution, it is shown that the proposed approach is much more exact and efficient compared to the known variants of the integral computational scheme, including the heat-balance integral method allowing the Stefan condition at the free boundary to be satisfied. The solutions obtained with the use of the square-law and cubic polynomials are presented. As for the test problem using the cubic polynomial, the relative error in determining the free boundary comprises hundredths and thousandths of percent. In this case, at the time instant t = 1, the relative error in determining the temperature profile is e T = 0.075 %.