Reaction boundary layer with square-root thickness ($ϵy^{\prime \prime }=y-f(x),X=(1-x)/\sqrt{ϵ}$)

For a positive small parameter $ϵ$, an order-one mismatch between the reduced solution $y_0=f(x)$ and a prescribed right endpoint value $y(1)=b$ creates a right boundary layer. Balancing $ϵy^{\prime \prime }$ against the order-one reaction term gives width $\sqrt{ϵ}$. In the inner variable $X=(1-x)/\sqrt{ϵ}$, the leading equation is $Y_0^{\prime \prime }=Y_0-f(1)$. Its boundary and matching conditions are $Y_0(0)=b$ and $Y_0\to f(1)$ as $X\to \infty$, soThe leading additive composite expansion is $f(x)+(b-f(1))e^{-(1-x)/\sqrt{ϵ}}$, obtained by subtracting the common overlap value. For the nonlinear perturbation $ϵy^{\prime \prime }=y-x^2+ϵ(2+y^2)$ with zero endpoint values, the first two outer expansion terms are $x^2-ϵx^4$, and the leading right-layer term is $1-e^{-X}$.