 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2015/paper-2/20/b/solution

The steady profile has no time derivative. Substituting

u = u + u_{s}

and using its steady equation cancels the forcing, giving

u_{t} = ν u_{xx}, u (0, t) = u (1, t) = 0 .

u (x, 0) = - u_{s} (x) = - \frac{G}{ν n ^{2} π ^{2}} sin (nπ x) .

The resulting heat equation is independent of

G

, as are its boundary conditions. Its initial amplitude still depends on

G

; removing a source from the evolution equation does not remove it from the whole problem.

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