Suppose I'm doing a Laplace Transform or Fourier Transform with respect to $t$ to solve a PDE, and I have Neumann (partial derivative) boundary conditions, such as $$u_t(a, t) = f(t)$$
Can I just translate these directly, such as $$U_t(a, s) = F(s)$$
or do I have to apply the same transform of derivative formulas used to transform partial derivatives in the PDE to the left hand side of the boundary condition equations, i.e.$$sU(a, s) - u(a, 0) = F(s)$$
or is some other transformation correct?