$$\frac{\partial u}{\partial t} = 2\frac{\partial^2 u}{\partial x^2}, \quad t > 0, 0 < x < 1,$$
$$\frac{\partial u}{\partial x}(0,t) = 0, \quad t > 0,$$
$$\frac{\partial u}{\partial x}(1,t) = 0, \quad t > 0,$$
$$u(x,0) = 1 - 2\cos(\pi x) + 2\cos(3\pi x), \quad 0 \le x \le 1.$$
Compute $\lim_{t \to \infty} u(x,t)$ and check that it equals the average value of $u(x,0)$.
