A rectangular lamina in the \( x y \)-plane with width \( a \) and height \( b \) is positioned between two ice slabs, making the left and right edges maintain a temperature of \( T=0 \). The top and bottom edges are thermally insulated. The coordinate system is set with the origin at the lamina's lower left corner, as shown in the figure below.
We model this situation by a temperature function \( T(x, y, t) \) that satisfies the two-dimensional diffusion equation with thermal diffusivity \( D \) coefficient
\[
\frac{\partial T}{\partial t}=D \nabla^{2} T, \quad 0<x<a, 0<y<b,
\]
together with Dirichlet boundary conditions on the left and right hand sides
\[
T(0, y, t)=0, \quad T(a, y, t)=0, \quad 0<y<b,
\]
and Neumann boundary conditions on the other two sides
\[
\frac{\partial T}{\partial y}(x, 0, t)=0, \quad \frac{\partial T}{\partial y}(x, b, t)=0, \quad 0<x<a .
\]
(a) Show that applying the method of separation of variables using \( T(x, y, t)=V(x, y) U(t) \) gives the solution \( U(t)=e^{\mu D t} \) and the eigenproblem \( \nabla^{2} V(x, y)=\mu V(x, y) \).
(b) Applying the method of separation of variables again, using \( V(x, y)=X(x) Y(y) \), gives the two further eigenproblems \( X^{\prime \prime}=(\mu-\lambda) X \) and \( Y^{\prime \prime}=\lambda Y \).
Translate the boundary conditions for \( T \) into boundary conditions for \( X(x) \) and \( Y(y) \).
(c) The \( Y(y) \) eigenproblem is analysed in Section 3.2 of the text, and the eigenfunctions are
\[
Y_{n}(y)=\cos \left(k_{n} y\right), \quad \text { where } k_{n}=n \pi / b, n=0,1,2, \ldots,
\]
with corresponding eigenvalues \( \lambda=-k_{n}^{2} \).
Find the eigenfunctions and eigenvalues for the \( X(x) \) eigenproblem.
(d) Use the eigenfunctions from part (c) to write down the eigenfunctions and eigenvalues for the \( V(x, y) \) eigenproblem.
(e) Write down a general solution for the temperature \( T(x, y, t) \).
