Solving 2D parabolic PDE

Question

Please forgive if this is simple, but I was wondering if one may be able to derive a closed-form solution to

\begin{align} \frac{\partial u}{\partial t} & = \frac{\partial^2 u}{\partial x^2} + \frac{1}{2}\frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 u}{\partial y^2}, \\ u(0,x,y) & = \sin(x)\sin(y) \\ u(t,x,y) & = 0, \qquad (x,y) \in \partial \Omega \end{align}

on $\Omega = [0,\pi] \times [0,\pi]$. I'm aware of series solutions for when there is no mixed derivative term using separation of variables, but is there a method of deriving solutions for these sorts of PDEs? I would appreciate a good push in the right direction!

Answer 1

You can make your PDE into the standard heat equation $u_t = u_{XX} + u_{YY}$ by introducing a linear change of variables $x = \sqrt{5/8} X + \sqrt{3/8} Y$, $y = \sqrt{5/8} X - \sqrt{3/8} Y$. Unfortunately, this messes up the boundary conditions: the square $[0,\pi] \times [0,\pi]$ becomes a parallelogram, and so the eigenfunctions won't have a nice product structure. Computing the eigenfunctions and eigenvalues of the Laplacian on a parallelogram seems to be not an easy problem: see e.g. this discussion on MathOverflow. There may not be a closed-form solution.