 Solution

On the two independent variables, the total derivative is $D_x={\partial }_x+{ϵ}^{-1}{\partial }_X$. Multiplication of $D_x(a{D}_{x}y)=f$ by ${ϵ}^2$ gives exactly the stated multiple-scale equation.

At order one, ${\partial }_X(a{\partial }_X{y}_0)=0$. Integrating twice givesThe periodic reciprocal coefficient has strictly positive mean. Unless $A=0$, the integral has a nonzero linear drift in $X$, violating the periodic cell ansatz and creating secular growth when $X=x/ϵ$. Thus $A=0$ and $y_0=y_0(x)$.

At order $ϵ$,Periodicity requires the derivative to have zero cell average. ConsequentlyAt order ${ϵ}^2$, the equation isThe second flux inside braces is $C(x)$, independent of $X$. Integrating over one period annihilates the first derivative, by periodicity, and yields $C^{\prime }(x)={\int }_0^{1}f(x,X)dX$. HenceThe effective transport coefficient is the harmonic mean of $a$ over the fast period; the source is averaged arithmetically. Smoothness sufficient for these formal differentiations and uniform positivity on the cells are understood.