Solve the heat equation
∂u/∂t = ∂²u/∂x², 0 < x < L, t > 0
∂u/∂x(0,t) = 0, ∂u/∂x(L,t) = 0, t > 0
u(x, 0) = x²
This models the heat equation in a long thin rod with insulated ends. We know that we can solve the PDE using the series solution:
u(x,t) = ∑[n=0 to ∞] (2/L)(∫[0 to L] u(x, 0) cos(πn/L) dx) cos(nπx/L) exp(-n²π²t/L²)
Remember that 2 has already been factored out.