A Revised Approach for One-Dimensional Time-Dependent Heat Conduction in a Slab

Classical Green's and Duhamel's integral formulas are enforced for the solution of one dimensional heat conduction in a slab, under general boundary conditions of the first kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.