 Solution

For $\sigma =-1$, the reduced first-order outer equation isThe condition at $x=0$ selectsIndeed $z_0\to 1$ as $x\to 0^{+}$, while $z_0\to 0$ as $x\to 1^{-}$. The boundary condition at $x=1$ must therefore be supplied by a boundary layer.

SetSince $x{\log }^{2}x=O({\epsilon }^2{X}^2)$ in this layer, the convection term is lower order. The leading inner equation and matching conditions areThusThe leading uniformly valid expression isNear $x=0$ diffusion regularizes the divergent derivative of the outer approximation on the thinner scale $x|\log x|=O(\epsilon )$, but the leading value there remains the already matched constant $1$.