Asymptotic Expansions for Two Singularly Perturbed Convection–Diffusion Problems with Discontinuous Data: The Quarter Plane and the Infinite Strip
We consider a singularly perturbed convection–diffusion equation, , defined on two domains: a quarter plane, (x, y) ∈ (0, ∞) × (0, ∞) , and an infinite strip, (x, y) ∈ (−∞, ∞) × (0, 1) . We consider for both problems discontinuous Dirichlet boundary conditions: u(x, 0) = 0 and u(0, y) = 1 for the first one and u(x, 0) =[a,b](x) and u(x, 1) = 0 for the second. For each problem, asymptotic expansions of the solution are obtained from an integral representation in two limits: (a) when the singular parameter → 0+ (with fixed distance r to the discontinuity points of the boundary condition) and (b) when that distance r→ 0+ (with fixed ). It is shown that in both problems, the first term of the expansion at = 0 is an error function or a combination of error functions. This term characterizes the effect of the discontinuities on the -behavior of the solution and its derivatives in the boundary or internal layers. On the other hand, near the discontinuities of the boundary condition, the solution u(x, y) of both problems is approximated by a linear function of the polar angle at the discontinuity points.
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Document Type: Research Article
Publication date: July 1, 2004