 Solution

ID: past-exam-of-the-year-2-of-the-earth-sciences-geology-course-of-the-university-of-oxford/2023/paper-1/10/b/ii/solution

For

T_{n} = A cos (k_{n} x) e^{- κ k_{n}^{2} t}

, direct differentiation again gives

\partial_{t} T_{n} = - κ k_{n}^{2} T_{n}

and

\partial_{x}^{2} T_{n} = - k_{n}^{2} T_{n}

. Thus it satisfies the heat equation. Moreover,

\partial_{x} T_{n} = - A k_{n} sin (k_{n} x) e^{- κ k_{n}^{2} t}

vanishes at both endpoints for integer

n

, proving the homogeneous Neumann boundary conditions. Now

n = 0

gives the nonzero constant mode

T = A

, which does not decay. It represents conserved mean temperature in an insulated plate.

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