A powerful method of analysis, the combined use of the Laplace transform and the finite-element method, is applicable to the problem of time-dependent heat flow systems. The present method removes the time terms using the Laplace transform and then solves the associated equation with the finite-element technique. The associated temperature transform is inverted by the method of Dubner and Abate for obtaining the required results. The present results are compared in tables to the analytical solutions and other numerical data, and it is found that the present solutions are stable and convergent to them. There exists no time step; thus, the present method is a useful tool in solving lengthy problems or in other engineering applications.