Analytic Approximations for a Singularly Perturbed Convection—Diffusion Problem with Discontinuous Data in a Half-Infinite Strip

Author： López José Pérez Sinusía Ester

Publisher： Springer Publishing Company

ISSN： 0167-8019

Source： Acta Applicandae Mathematicae, Vol.82, Iss.1, 2004-05, pp. : 101-117

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Abstract

We consider a singularly perturbed convection—diffusion equation, −∈∆u+v⋅∇ u=0, defined on a half-infinite strip, (x,y)∈(0,∞)×(0,1) with a discontinuous Dirichlet boundary condition: u(x,0)=1, u(x,1)=u(0,y)=0. Asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) as the singular parameter ∈→0⁺ (with fixed distance r to the discontinuity point of the boundary condition) and (b) as that distance r→0⁺ (with fixed ∈). It is shown that the first term of the expansion at ∈=0 contains an error function or a combination of error functions. This term characterizes the effect of discontinuities on the ∈-behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the point of discontinuity of the boundary condition, the solution u(x,y) is approximated by a linear function of the polar angle at the point of discontinuity (0,0).