Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2010 / A / 1

Put $\delta =\sqrt{ϵ}$ and write the outer asymptotic expansion as $y_0+\delta y_{1/2}+{\delta }^2{y}_1+\cdots$. Equating powers in the differential equation givesThe first two equations give $y_0=A{e}^{x^2/2}$ and $y_{1/2}=B{e}^{x^2/2}$. Since $y_0^{(3)}=A(x^3+3x)e^{x^2/2}$, an integrating factor yieldsHere $A,B,C$ are matching constants; boundary conditions have not yet been imposed.

In a layer of thickness $\ell \ll 1$, the third-derivative and first-derivative terms have sizes $ϵ{\ell }^{-3}y$ and ${\ell }^{-1}y$. Their dominant balance requires ${\ell }^2\sim ϵ$, soThe local fast modes are exponentials with rates $\pm 1/\sqrt{ϵ}$. A mode decaying into the interval can occur at either endpoint: $e^{-x/\sqrt{ϵ}}$ at zero and $e^{-(1-x)/\sqrt{ϵ}}$ at one. The zero fast exponent represents the matching outer component. The term $xy$ is lower order on either fast scale, and even smaller near zero; it creates no further independent layer thickness.

For the left inner coordinate $X=x/\delta$, the equation becomes $Y_{XXX}-Y_X+{\delta }^{2}XY=0$. At leading order the solution that matches a bounded outer expansion is $Y_0=A+D{e}^{-X}$; the growing exponential is excluded. The boundary value gives $A+D=1$. Since $y^{\prime }(0)=1$ is finite, the possible $O({\delta }^{-1})$ derivative must vanish, so $D=0$ and $A=1$.

At the next order, $Y_{1/2}=B+E{e}^{-X}$. Its boundary value is zero, giving $E=-B$, while its physical derivative at zero is $-E=1$. Hence $B=1$, and the left inner expansion is $1+\delta (1-e^{-X})+O({\delta }^2)$.

At the right endpoint put $\xi =(1-x)/\delta$. The equation is $Y_{\xi \xi \xi }-Y_{\xi }=\delta (1-\delta \xi )Y$. Matching the outer expansion at one and imposing $Y(0)=1$ giveAt the next order, $Y_{1/2}^{(3)}-Y_{1/2}^{\prime }=Y_0$. The solution with no growing exponential, the correct matching polynomial, and $Y_{1/2}(0)=0$ isIndeed the operator $D^3-D$ sends $-\sqrt{e}\xi$ to $\sqrt{e}$ and sends $\xi e^{-\xi }$ to $2e^{-\xi }$.

Add the two decaying inner corrections to the outer solution, subtracting their common matching parts. Since $\delta \xi =1-x$, the composite asymptotic expansion isIt reproduces all three boundary conditions to the retained algebraic orders, with exponentially small interactions between endpoints. The right endpoint has an $O(1)$ boundary layer in $y$. The left correction has only amplitude $O(\sqrt{ϵ})$ in $y$, but its derivative is $O(1)$ and is necessary for the derivative boundary condition. Thus the left layer is a weak derivative boundary layer; if “genuine” refers specifically to an order-one change in the function itself, only the right layer is genuine in that sense. Both nonzero corrections belong in the expansion through $O(\sqrt{ϵ})$.