$\partial_t u= k\partial^2_x u , 0<x<\pi, t>0 \\ u(x,0)=cos^2 (x) + sin^2 (3x) \\ \partial_xu(0,t)=\partial_x u(\pi,t)=0 $
This is a PDE I am given to solve. My question is to just verify the follwing: Where is the initial condition stated used to solve this Neumann Heat Equation?
I know that:
$u(x,t)= 1/2 A_0 + \sum A_n e^{-(n\pi/l)^2kt}cos(n\pi x/l)\\ \phi(x)= 1/2 A_0 + \sum A_n cos(n\pi x/l) $.
Edit: Looking at the question a bit closer, is the problem just asking to solve for the unknown constants given $\phi(x)$?
Notice that you have $$\phi(x) = \cos^2(x)+sin^2(2x) = u(x,0) = \frac{1}{2}A_0+\sum_{n=1}^\infty A_n\cos(n\pi x/l).$$ I'm assuming that the problem should be posed for $0<x<l$, otherwise I cannot see where the $l$ comes from.
Integrate this equation, and you obtain $A_0 = 2\int_0^l\phi(x)$. To obtain the rest of the $A_n's$ simply multiply the equation by $\cos(m\pi x/l)$ and integrate from $0$ to $l$, utilising the orthogonality of the set of functions $\{\cos(m\pi x/l)\}_{m=0}^\infty$.