Short-time Asymptotic Solutions of the Heat Conduction Equation with Spatially Varying Coefficients
George R. Gavalas, Yanis C. Yortsos · IMA Journal of Applied Mathematics · 1980
An asymptotic solution of the heat conduction equation with spatially varying coefficients is developed for small times. The method followed consists of an application of the Laplace transformation and use of the Liouville-Green approximation in the subdominant solution of the resulting second-order differential equation. The approximate solution is inverted by contour integration. The resulting asymptotic expression has a time-dependence identical to that applying to the case of constant properties provided that an appropriately averaged value of the thermal diffusivity a is used, namely, d* The error term is of the order of t||^| | where ||^| | is a measure of spatial variability of the coefficients in the heat equation. 1.