Particular solutions of singularly perturbed partial differential equations with constant coefficients in rectangular domains, part I. convergence analysis

  • Authors:

  • Zi-Cai Li;Hsin-Yun Hu;Chung-Hua Hsu;Song Wang

  • Affiliations:

  • Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan;Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan;Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan;Department of Mathematics & Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia

  • Venue:
  • Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on boundary and interior layers - computational and asymptotic methods (BAIL 2002)
  • Year:
  • 2004

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Abstract

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 ε (e.g., ε = 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 small ε (e.g., ε = 10-7), but Model I for a moderately small ε (=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.