Solve heat equation using separation of variables

Question

I am trying to solve the following exercise:

$u_t=4u_{xx}, 0,x<x<\pi, t>0$

$u(x,0)=g(x), 0\le x\le\pi$

$-u_x(0,t)=u_x(\pi,t)=0, t>0$,

considering that $u(x,t) \to U$ as $t \to \pm \infty$.

I tried so solve it using separation of variables (suppose $u(x,t)=X(x)T(t)$) and using the Neumann boundary conditions and the initial condition in order to find the constants. I came up with the following:

$u(x,t)= \sum_{k=1}^ \infty a_k \cos (kx) e^{-4k^2t}$

$a_0 = \frac{1}{\pi} \int_0^\pi g(x)dx$

$a_k = \frac{2}{\pi} \int_o^\pi f(x) \cos(kx)dx$.

Is this correct? What does it mean that $u(x,t) \to U$ as $t \to \pm \infty$?

EDIT:

From my solution $u(x,t)$ I use the initial condition and get:

$g(x) = \sum_{k=1}^\infty a_k \cos(kx)$.

From this I should use Fourier series to find $a_k$ right?

Answer 1

I think this is the correct procedure/answer:

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