Resumen de The role of the error function in some simple singular perturbation problems
Juan Carlos Jorge Ulecia, José Luis López García
Two singular perturbation problems in two dimensions are analyzed: a) convection-diffusion in a quarter plane and b) convection-diffusion in an infinite strip. In both cases we consider constant coefficients and certain discontinuous Dirichlet boundary conditions. For each problem, asymptotic expansions of the solution are obtained from an integral representation in two limits: a) when the singular parameter goes to zero (with fixed distance r to the discontinuity point of the boundary condition) and b) when that distance r goes to zero (with fixed ). All the expansions are accompanied by error bounds at any order of the approximation. The asymptotic expansion at = 0 (valid for r ≥ r0 > 0) is derived by a classical method, whereas the derivation of the expansion at r = 0 (valid for ≥ 0 > 0) requires the distributional approach applied on Laplace transforms. It is shown that, for → 0, the first term of the solution’s expansion is an error function or a combination of error functions. This term characterizes the effect of the discontinuities on the behavior of the solutions and their derivatives with respect to in the boundary layers. The behavior of the solution near the discontinuities is described by the expansion at r = 0. A careful look to these expansions give the keys to develop robust and efficient numerical methods (uniformly convergent with respect to ) for the resolution of convection-diffusion problems with discontinuities in the inflow boundary.
