Get rid of mixed derivatives in stationary PDE using coordinate transform

Question

In this question the PDE $$u_{xx} + u_{yy} + u_{zz} + u_{zy} = 0,$$ was rewritten to the standard Laplace equation $\Delta u=0$ using a coordinate transform, in some sense similar how you can map an ellipse to a circle and vice versa. I was wondering if a similar technique can be applied to the PDE $$-u_{xx} - u_{xy} - u_{yy} + 5u = f.$$ Possibly with Neumann boundary conditions.

Answer 1

You can make change of variables: $\xi = y - \frac{x}2$, $\eta = \frac{x\sqrt{3}}2$, $v(\xi, \eta) = u(x, y)$. And you will get an equation $$ 5 v - \frac{3}4 \partial_\xi^2 v- \frac{3}4 \partial_\eta^2v = f, $$ or $$ \left(\Delta - \frac{20}3\right)v(\xi, \eta)=-\frac{4}3f(\xi, \eta). $$ It is inhomogeneous Helmholtz equation. Such equations are studied separately from the Poisson and Laplace equations, but in a similar way.