 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2017/paper-1/2/c/solution

The nonstationary part of

u

decays exponentially, whereas the integrating factor grows quadratically. For fixed

x > 0

\frac{v ( x , t )}{3 x t ^{2} / π} = (1 + \frac{1}{t})^{2} (1 + \frac{e ^{- 4 t} s i n 2 x}{3 x}) ⟶ 1.

Consequently

v (x, t) \sim \frac{3 x t ^{2}}{π} (t \to \infty) .

x = 0

, both expressions are identically zero; the ratio formulation is understood for

x > 0

. The diffusion modes have died away, leaving the lifted linear profile multiplied by the reaction-induced amplitude factor.

 New to topics? Read the docs here!