🚀 go-pugleaf

Good day,
   I'm interested in solving a Diffusion equation problem using FEM.  However,
I'm having difficulty putting the equation into a functional form that can be
manipulated.  The equation is a 1-D equation with the following boundary
conditions:
       u(x,0) = f(x) = sin x
and    u(0,t) = u(L,t) = 0  as bc's.
the actual equation as such comes to a solution of
   u(x,t) = sin (x) (e**(-k**2 t )
when solved analytically.
When attempting to use the Euler-Lagrange eqn. to find the functional form,
I was having extreme difficulties in determining my course of action,
particularly in solving for adjoint operators in the equation.
Any help would be appreciated.
 
Ut = k**2 Uxx is the Diffusion equation.

Thanx.
A.C.