||In the …rst chapter, separation of variables is used to derive the explicit particular solutions for a class of singularly perturbed di¤erential equations with constant coe¢ cients on a rectangular domain. Although only the Dirichlet boundary condition is taken into account; it can be similarly extended to other boundary conditions. Based on these results, the behavior of the solutions and their derivatives can be easily illustrated. Moreover, we have proposed a model with exact solution, which can be used to explore the behavior of layer and to test numerical methods. Hence, these analytic solutions are very important to the study in this …eld. In the second chapter, we study the model of Shi¤ et al. . It is a biharmonic equation on the rectangular domain [¡ a; a]£ [0; b] with clamped boundary condition. We compute its most accurate numerical solution by boundary approximation method (BAM), which is a special version of spectral method or collocation method. Its convergence unfortunately is not as good as the usual spectral method with exponential decay rate. We discover that the slowdown is due to the very mild singularity at two corners not considered by BAM. We further simplify the basis functions and their partial|
derivatives. Using these functions we can construct several models useful for testing numerical methods. We also explore how the stress intensity factor depends on the sizes of domain a and b, and the load ¸ by reducing the original problem with three parameters lambda, a, b to that with only one parameter t.