The technique of separation of variables is used to derive explicit particular solutions for constant coefficient, singularly perturbed partial differential equations (PDEs) on a rectangular domain with Dirichlet boundary conditions. Particular solutions and exact solutions in closed form are obtained. An analysis of convergence for the series solutions is performed, which is useful in numerical solution of singularly perturbed differential equations for moderately small values of epsilon(e.g., epsilon = 0.1-10(-4)). Two computational models are designed deliberately: Model I with waterfalls solutions and Model II with wedding-gauze solutions. Model II is valid for very epsilon (e.g., epsilon = 10(-7)), but Model I for a moderately small epsilon (=0.1-10(-4)). The investigation contains two parts. The first part, reported in the present paper, focuses on the convergence analysis and some preliminary numerical experiments for both of the models, while the second part, to be reported in a forthcoming paper, will illustrate the solutions near the boundary layers. (C) 2003 Elsevier BY. All rights reserved.