A distributional approach to the boundary layer theory
The aim of this paper is to show that boundary layer theory for linear problems can be investigated using a distributional approach. As a first problem, the authors consider the solution \(y(x, \varepsilon)\) of the initial value problem \[ \varepsilon y^{(m)} + \sum^{m - 1}_{j = 0} a_j (x)y^{(j)} = f(x), \quad y^{(j)} (0) = y_j,\;j = 0, \ldots, m - 1, \] where \(\varepsilon\) is a small parameter. Under natural assumptions, they prove an expansion \[ z(x, \varepsilon) = H(x) y(x, \varepsilon) \sim \sum^\infty_{n = 0} z_n(x) \varepsilon^n , \] where \(H(x)\) is the Heavyside function and the \(z_n\) are in a distribution space. A second result refers to the possibility to split the expansion of \(z(x, \varepsilon) \sim z^{(ou)} (x, \varepsilon) + z^{(in)} (x, \varepsilon)\) into an outer and an inner expansion \[ z^{(ou)} (x, \varepsilon) \sim \sum^\infty_{n = 0} z_n^{(ou)} (x) \varepsilon^n, \quad z^{(in)} (x, \varepsilon) \sim \sum^\infty_{n = 0} z_n^{(in)} (x) \varepsilon^n, \] where the \(z_n^{(ou)}\) are regular and the \(z_n^{(in)}\) with support in \{0\}. A second problem concerns the boundary value problem \(\varepsilon y'' + a(x)y' + b(x)y = f(x)\), \(y(0) = A\), \(y(1) = B\). Introducing the auxiliary quantity \(C (\varepsilon) : = y' (0, \varepsilon)\), the authors obtain expansions \[ z(x, \varepsilon) = H(x) y(x, \varepsilon) \sim \sum^\infty_{n = 0} z_n (x) \varepsilon^n , \quad C (\varepsilon) \sim \sum^\infty_{n = - 1} C_n \varepsilon^n, \] where the distributions \(z_n\) are functions of \(C\). All these results are illustrated with various examples which clarify the scope of the abstract results.
scientific article; zbMATH DE number 811163