C3.
(a) The function $u(x, t)$, defined over the interval $0 \le x \le 1$ for $t \ge 0$, satisfies the heat equation
$u_t = u_{xx}$, subject to boundary conditions $u(0, t) = 0$, $u(1, t) = 0$. You may assume that $u$ is not
the trivial solution, i.e. $u(x, t) \ne 0$ for some $x$ and $t$.
i) Using separation of variables, write $u(x, t) = X(x)T(t)$ and determine all pairs of admissible
functions $X(x)$ and $T(t)$. (You do not need to specify the amplitudes of these functions.)
ii) Use linear superposition to write down the general solution $u(x, t)$.
[5 marks]
[2 marks]
iii) Assuming that $u(x, t)$ satisfies the initial condition $u(x, 0) = \sin(\pi x)$, show that $u(\frac{1}{2}, t) = e^{-\pi^2 t}$.
[2 marks]
