5. Consider the initial-boundary value problem for $u(x, t)$ which satisfies
$\nabla^4 u + \frac{\partial u}{\partial t} = \phi(x, t);$
$x \in \Omega \quad 0 < t < \infty$
with the boundary and initial conditions $u = f(x, t)$ and $\frac{\partial u}{\partial n} = h(x, t)$ for $x \in \partial \Omega$
and the initial condition $u(x, 0) = F(x)$ for $x \in \Omega$. Using the Green's function
$g(x, t \mid x_0, t_0)$, where
$\nabla^4 g + \frac{\partial g}{\partial t} = -\delta(x - x_0)\delta(t - t_0),$
with appropriate boundary conditions, determine an explicit solution for this
problem paralleling the arguments we did in class for the heat equation. The
answer will depend on the initial and boundary data and the Green's function.
All the details must be clearly given.
