A thin slab of metal is formed in the shape of a square. The \(x-y\) vertices of the square are \((0,0),(1,0),(0,1)\), and \((1,1)\). The faces of the slab in the \(z\)-direction are insulated so that all heat transfer occurs in the \(x y\)-plane. The initial temperature distribution in the plane is \(T(x, y, 0)=T_{0}(x, y)\). Assuming there are no sources of heat in the slab, that the \(x\)-faces are insulated and that the \(y\)-faces are held at a temperature \(T=0\) :

a. Derive the generic differential equation describing the situation.

b. Show that the equation:

\[T_{m n}(x, y, t)=G_{m n} \sin (m \pi x) \cos (n \pi y) \exp \left[-\alpha\left(m^{2}+n^{2}\right) \pi^{2} t\right]\]

satisfies the differential equation.

c. A single value of \(G_{m n}\) cannot represent an arbitrary initial condition so we resort to using a sum. In this case, the \(G_{m n}\) are given by:

\[G_{m n}=4 \int_{0}^{1} \int_{0}^{1} T_{0}(x, y) \cos (n \pi y) \sin (m \pi x) d y d x\]

If \(T_{0}(x, y)=x y\), determine the full solution for \(T\).