By Ferdinand Verhulst
Includes well-chosen examples and exercises
A student-friendly creation that follows a workbook kind technique
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Additional resources for Methods and Applications of Singular Perturbations
1. Domain D with boundary layer near S. 2, is called a boundary layer. We now characterise such regions. 1 The Concept of a Boundary Layer Consider the function φε (x) deﬁned on D ⊆ Rn . Suppose there exists a connected subset S ⊂ D of dimension ≤ n, with the property that φε (x) has no regular expansion in each subset of D containing points of S (see Fig. 1). Then a neighbourhood of S in D with a size to be determined, will be called a boundary layer of the function φε (x). 2 the domain is one-dimensional.
There is no way to satisfy the boundary conditions with the regular expansion. The exact solution can be written as φε (x) = e−x/ε + 0(e−1/ε ), which shows that the behaviour of the solutions is diﬀerent in two subdomains of [0, 1]: in a small region of size 0(ε) near x = 0, the solution decreases very rapidly from 1 towards 0; in the remaining part of [0, 1], the solution is very near 0. The regular expansion is valid here with, rather trivially, φn (x) = 0, n = 0, 1, · · · . Note that even in the domain where the regular expansion is valid, the choice to expand with respect to order functions of the form εn is not a fortunate one as e−1/ε = o(εn ), n = 1, 2, · · · .
2 A Boundary Layer at One Endpoint We consider again the boundary value problem formulated at the beginning of this chapter, εL1 φ + L0 φ = f (x), φε (0) = α, φε (1) = β, but now with L0 = b1 (x) d + b0 (x). dx As before, we have L1 = a2 (x) d2 d + a0 (x) + a1 (x) dx2 dx with a2 (x) > 0, x ∈ [0, 1]; all coeﬃcients are assumed to be suﬃciently diﬀerentiable. Suppose, moreover, that b1 (x) does not change sign, say b1 (x) < 0, x ∈ [0, 1]. The case in which b1 (x) vanishes in the interior of the interval is called a turning-point problem.
Methods and Applications of Singular Perturbations by Ferdinand Verhulst