Approximate analyses of Fourier and non-Fourier heat conduction models by the variational principles based on Laplace transforms

Approximate analysis is a major application of variational principles for heat conduction. Recently, O'Toole's variational principle for Fourier's law has been extended to non-Fourier heat conduction models, which are applied to approximate analyses based on the Rayleigh-Ritz method. Suitable trial functions satisfying boundary conditions are sought, and then substituted into the variational principles to obtain the undetermined coefficients. From the inverse Laplace transforms, the approximate solutions are obtained. Examples are provided for 1D problems for different heat conduction models. The largest calculation errors are one or two orders of magnitude smaller than the equilibrium temperature, which will tend to be zero.