Solving PDE with peridoc boundary conditions and a sine as initial condition

Question

I need some help this exercise:

I have to solve diffusion equation $u_t = Du_{xx}$ with periodic boundary conditions

$u(t,-l) = u(t,l)$ and $u_x(t,-l) = u_x(t,l) $.

The initial value should be a function $f(x) = H \sin(2\pi\nu x)$, and $l=\pi$.

I have two questions:

1. Is it possible to solve this problem?
2. When,yes - How to solve it?

Thank you for your help in advance.

Answer 1

You could solve this problem using the method of separation of variables.

Then the solution has the form: $u(t,x)=X(x)T(t)$.

Replace this at the problem, and then you have to solve two problems, one of $X$ and one of $T$.