Improved approximate methods for analysing steady heat conduction

An improved version of the Rayleigh technique used by Rayleigh and Schmidt for solid mechanics problems is extended to oneand two-dimensional problems of steady heat conduction. Here the technique is applied to the heat-balance integral (HBI) method, the variational method and the square-error method. To determine the most appropriate value of the non-integer exponent, either one of two different criteria is used: the variational or the square-error integral. Thus, four new methods are considered. A comparison with a closed-form solution is made for a one-dimensional fin. It is demonstrated that the error for a one-term trial function is much less for an improved HBI method with the variational integral criterion. Thus, additional terms should not be needed in most instances. Excellent results are also achieved by using one of the new methods in a two-dimensional problem with internal heat generation.