Optimizing the exponent in the heat balance and refined integral methods

The most contentious aspect of the Heat Balance Integral Method is the choice of power of the highest order term in the approximating function. In this paper we develop a new method, where the exponent is determined during the solution process. This is achieved by minimising the square of the difference of the terms in the heat equation. The solution requires no knowledge of an exact solution and generally produces significantly better results than previous models. The method is illustrated by applying it to three standard thermal problems.