 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-2/20/a/solution

Θ_{t} = κ Θ_{xx}

, separation of variables with homogeneous Neumann boundary condition at both ends gives spatial modes

cos (nπ x / L)

and decay factors

e^{- κ (nπ / L)^{2} t}

. The

n = 0

mode is constant and does not decay.

The initial profile already contains only two such modes:

T_{0} sin^{2} \frac{π x}{L} = \frac{T _{0}}{2} - \frac{T _{0}}{2} cos \frac{2 π x}{L} .

Evolving each mode therefore yields

Θ (x, t) = \frac{T _{0}}{2} [1 - e^{- 4 κ π^{2} t / L^{2}} cos \frac{2 π x}{L}] .

The endpoint derivatives vanish, and substitution checks both the equation and the initial profile. The spatial integral remains

L T_{0} /2

: no heat escapes through insulated ends, so the limiting uniform temperature is

T_{0} /2

Insulated and zero-temperature endpoints evolve the same initial heat profile differently

. Original mode calculations. Both panels start from the same squared sine. Insulated endpoints conserve the mean temperature and approach

T_{0} /2

; zero-temperature endpoints remove heat and the profile decays to zero. Times are expressed as

τ = κ π^{2} t / L^{2}

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