Consider the equation $u_t = 16 u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t) = 0 , u(1,t) = 0$. Suppose
$\displaystyle u(x,0) = \sin(\pi x) + \sum_{n=1}^\infty \frac{3}{n^{3}} \sin(2 n \pi x) .$

If $u(x,t)$ represents the temperature of a rod at a position $x$ and time $t$, then at time $t$ the midpoint has the temperature

$\displaystyle u\Big(\frac{1}{2},t\Big) = {}$
help (formulas)