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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2024

Paper 3

Open ends impose $C(0,t)=C(L,t)=0$. By <separation of variables>,
$$
C(x,t)=\sum_{n=1}^\infty b_n\sin\frac{n\pi x}{L}
\exp\left[\left(\mu-\frac{Dn^2\pi^2}{L^2}\right)t\right].
$$
Expanding the initial constant gives $b_n=2C_0(1-(-1)^n)/(n\pi)$, hence
$$
C(x,t)=\frac{4C_0}{\pi}\sum_{\substack{n\geq1\\n\ {
m odd}}}
\frac1n\sin\frac{n\pi x}{L}
\exp\left[\left(\mu-\frac{Dn^2\pi^2}{L^2}\right)t\right].
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A / a / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A / a / i / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 13A / a / i / Solution

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