Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 336 / 2

Use the fast time $t$ and slow time $T={ϵ}^{2}t$, and writeThere is no $O(ϵ)$ resonant forcing. Solvinggives a convenient particular solutionAt $O({ϵ}^2)$, the coefficient of $e^{it}$ in the forcing must vanish by the solvability condition in the method of multiple scales. This givesorThe two slow exponents satisfyThe second instability tongue of a weak Mathieu oscillator has real $s$ when $-1/12<f<5/12$. At either endpoint the repeated zero exponent permits a linearly growing slow solution, so boundedness for every initial condition requires the strict stable ranges

Let $X=x/ϵ$, $a(X)=1/d(X)$, and seekwith periodic correctors. The leading cell equation makes $u_0$ independent of $X$. At the next order, continuity of microscopic heat flux givesAveraging over one period and using $⟨u_{1X}⟩=0$ yieldsThus the periodic homogenization of a diffusion equation has effective diffusivity $1/\overline{d}$:

For the sawtooth profile, the two triangular areas giveThe homogenized problem is therefore the unit-diffusivity heat equation. Its half-line step solution isIt has the required initial and boundary limits, and direct differentiation verifies $u_t=u_{xx}$.