The heat-balance integral: 2. Parabolic profile with a variable exponent: The concept, analysis and numerical experiments

A heat-balance integral method employing a parabolic profile with a variable (self-adapting) exponent has been developed. The parabolic profile satisfies both the boundary conditions and the heat-balance integral at any value of the exponent, while the optimal solution requires minimization of the mean-squared error defined by the domain parabolic equation. The concept of a variable exponent results in simple relationships involving the LambertW function and the initial values defined through the profile calibration at x=0. Two simple 1-D problems with classic solutions are developed to demonstrate the method.