  We obtain solution representation formulas for some linear initial boundary
value problems posed on the half space that involve mixed spatial derivative
terms via the unified transform method (UTM), also known as the Fokas method.
We first implement the method on the second order parabolic PDEs; in this case
one can alternatively eliminate the mixed derivatives by a linear change of
variables. Then, we employ the method to biharmonic problems, where it is not
possible to eliminate the cross term via a linear change of variables. A basic
ingredient of the UTM is the use of certain invariant maps. It is shown here
that these maps are well-defined provided that certain analyticity issues are
appropriately addressed.
