The solution of transient diffusion in a composite material consisting of several different materials involves solving the Sturm-Liouville problem with discontinuous coefficients. An efficient numerical method and algorithm are presented in this paper for finding eigenvalues, eigenfunctions, and the related coefficients. The method is straightforward and powerful. It can be applied to solve transient diffusion problems in various geometries with any number of layers of different materials. This is a considerable advantage over the commonly used Laplace transform method, which becomes tedious when applied to composite materials consisting of more than two layers of different materials.