A reduction principle for singular perturbation problems
Consider singularly perturbed systems of the type \[ \begin{multlined} f(x^{(m)}, x^{(m-1)}, \dots ,x,y^{(n)}, y^{(n-1)}, \dots ,y)=0,\\ y^{(r)} + \varepsilon g(y^{(p)} ,\dots ,y,x^{(q)}, \dots ,x) + h(y^{(r-1)}, \dots ,y,x^{(s)} ,\dots ,x)=0,\end{multlined} \tag{*} \] with \(p >r, q>s\). The solution to \((*)\) satisfying initial or boundary conditions consists of a regular part (outer solution) and a boundary layer correction. The authors propose an algorithm to construct the outer solution to \((*)\) and apply it to some examples. There is no mathematical justification of this approach. The remarks of the authors concerning critical (degenerate) cases are given without any justifications. The references contain no recent result on singularly perturbed problems.
scientific article; zbMATH DE number 1311071