Solving a differential equation with Neumann boundary condition

How to solve
$$\frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2}$$
with the boundary conditions of
$$u(x,t) = \begin{cases} 0, & x=l \\ 0, & x=0, t=0 \\ A, & x=0, t>0 \end{cases}$$

And one more question: is the third condition is still considered Newumann or it is Dirichlet?