Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 50 / 3

A singular perturbation is one for which setting the parameter to zero does not give a regular approximation uniformly over the region of interest. The limiting equation may lose a derivative or boundary condition, or a formally small correction may accumulate over a long time. Here the last mechanism occurs.

At fixed $t$, substitute a regular asymptotic expansion. The leading equations and initial conditions give $y_0=3$ and $x_0=\sin t$. At first order,with zero initial corrections. ThereforeThe secular terms become comparable with the leading terms at $t=O({ϵ}^{-1})$. The fixed-time expansion is not uniform on the slow relaxation time.

Introduce $\tau =ϵt$ through the method of multiple scales, and write $x=A(\tau )\sin t+O(ϵ)$, $y=Y(\tau )+O(ϵ)$. Average the second equation over the rapid oscillation. Since the sine has zero mean, $Y^{\prime }=1-Y$, where the prime now denotes $d/d\tau$. With $Y(0)=3$, $Y=1+2e^{-\tau }$.

In the oscillator equation, $2(y^{\prime }{)}^2$ is of order ${ϵ}^2$. The order-$ϵ$ resonant cosine forcing is $-(2A^{\prime }+3YA)\cos t$. A bounded first correction requires its coefficient to vanish, soIntegration gives the slow oscillator damping by a relaxing auxiliary variable:Hence the uniform leading solution isTo check the uniform ordering, the instantaneous sine forcing in the auxiliary equation is absorbed by a bounded first correction $-ϵA(\tau )\cos t$, plus a bounded slow correction chosen to match the initial data. The resonant first-order oscillator forcing has already been removed. Remaining order-${ϵ}^2$ terms accumulate by at most $O(ϵ)$ over $0\le t\le K/ϵ$ for fixed $K$. The slow coefficients stay bounded on this interval, and variation of constants for the oscillator and the auxiliary first-order equation gives the displayed uniform errors. Expanding the amplitude near $\tau =0$ gives $A=1-9\tau /2+\cdots$, recovering the secular term of the failed regular expansion.

For the positive-drift equation, the outer expansion obeys $(1+x)y_0^{\prime }+y_0=0$. Select the boundary condition at the right endpoint, obtaining $y_0=2/(1+x)$. The first correction satisfiesIntegrating $[(1+x)y_1{]}^{\prime }=-4/(1+x{)}^3$ givesThe boundary condition mismatch is at $x=0$. Set $X=x/ϵ$ and use the inner expansion $Y=Y_0+ϵY_1+\cdots$. The inner equation is $Y_{XX}+(1+ϵX)Y_X+ϵY=0$. At leading order, matching to $2$ and imposing $Y(0)=1$ give $Y_0=2-e^{-X}$. At first order,The outer expansion in the overlap is $2+ϵ(3/2-2X)$. Matching its constant and linear terms givesThe additive composite expansion adds inner and outer approximations and subtracts the common overlap:It meets the left condition exactly and the right condition up to exponentially small terms, with an $O({ϵ}^2)$ uniform asymptotic error. This is the first-order composite expansion for a positive drift specialized to the present coefficients.

Reversing the drift moves the decaying boundary layer to $x=1$; the outer solution is selected at $x=0$. For the second equation, $y_{\mathrm{out}}=1+x$ is in fact an exact solution, but predicts $2$ at the right boundary instead of $1+ϵ$. Put $X=(1-x)/ϵ$. The inner equation becomesMatching to $2-ϵX$ givesIndeed, $Y_1^{\prime \prime }+2Y_1^{\prime }=-2+(1+2X)e^{-2X}$ and $Y_1(0)=1$, as required by the order-$ϵ$ boundary value. Since $d/dx=-{ϵ}^{-1}d/dX$, $Y_0^{\prime }(0)=2$ and $Y_1^{\prime }(0)=-4$ give the endpoint derivative in a negative-drift boundary layer:The order-one term cannot be obtained by differentiating the leading inner solution alone.