Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Asymptotic approximation of singularly perturbed convection-diffusion problems with discontinuous derivatives of the Dirichlet data

Authors: José L. López and Ester Pérez Sinusía
Journal: Quart. Appl. Math. 63 (2005), 527-543
MSC (2000): Primary 35C20, 41A60
Published electronically: August 17, 2005
MathSciNet review: 2169032
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Abstract: We consider a singularly perturbed convection-diffusion equation, $-\epsilon\bigtriangleup u +\overrightarrow v\cdot\overrightarrow\nabla u=0$, defined on two domains: a quarter plane, $(x,y)\in(0,\infty)\times(0,\infty)$, and a half plane, $(x,y)\in(-\infty,\infty)\times(0,\infty)$. We consider for these problems Dirichlet boundary conditions with discontinuous derivatives at some points of the boundary. We obtain for each problem an exact representation of the solution in the form of an integral. From this integral we derive an asymptotic expansion of the solution when the singular parameter $\epsilon\to 0^+$ (with fixed distance $r$ to the points of discontinuity of the boundary condition). It is shown that, in both problems, the first term of the expansion contains the primitive of an error function. This term characterizes the effect of the discontinuities on the $\epsilon-$behaviour of the solution and its derivatives in the boundary or internal layers.

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Additional Information

José L. López
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain
Email: jl.lopez@unavarra.es

Ester Pérez Sinusía
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006-Pamplona, Spain
Email: ester.perez@unavarra.es

DOI: https://doi.org/10.1090/S0033-569X-05-00962-0
Keywords: Singular perturbation problem, discontinuous boundary data, asymptotic expansion, error function
Received by editor(s): October 28, 2004
Published electronically: August 17, 2005
Article copyright: © Copyright 2005 Brown University

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