Consider the equation $u_t = u_{xx}$, $0 < x < 1$, $t > 0$, with boundary conditions $u(0,t) = 0 , u(1,t) = 0$. Suppose
$\displaystyle u(x,0) = \sum_{n=1}^\infty \frac{-6}{n^{7}} \sin(n \pi x) .$

Then the solution is

$\displaystyle u(x,t) = \sum_{n=1}^\infty$ $e^{-(n\pi)^2t} \sin(n\pi x)$

help (formulas)