Seperating variables for the homogeneous version of this equation leads to the same eigenvalue problem as in PDE7. Those eigenvalues and eigenfunctions are:
Now, represent the function in its eigenfunction expansion.
Let in the formula for above and use the inital condition to get
D is the rectangular domain , and C is its boundary.
To evaluate the coefficients, take the derivative of the last equation with respect to .
Substitute the DE for u
Label the second integral and save it for later. This is a known function of t.
Using Green's identities,
and the fact that , the first integral can be transformed:
The first integral on the right is equal to . The last integral around the curve is zero because the outward normal vector of u and φ are zero because both functions are zero there by the boundary conditions. And so
So is determined and gives the coefficient in the Fourier expansion of
The problem is formally solved. But just for fun, the explicit solution is
Which can probably be simplified.