Short-time Asymptotic Solutions of the Heat Conduction Equation with Spatially Varying Coefficients

An asymptotic solution of the heat conduction equation withspatially varying coefficients is developed for small times.The method followed consists of an application of the Laplacetransformation and use of the Liouville-Green approximationin the subdominant solution of the resulting second-order differentialequation. The approximate solution is inverted by contour integration.The resulting asymptotic expression has a time-dependence identicalto that applying to the case of constant properties providedthat an appropriately averaged value of the thermal diffusivity is used, namely, The error term is of the order of t where is a measure of spatialvariability of the coefficients in the heat equation.