Question: Consider $u_t = k(u_{xx} + u_{yy})$, x, y > 0 subject to the boundary conditions
u(0, y, t) = 0 and $u_y(x,0,t) = 0$ and initial condtion
u(x, y, 0) = f(x, y).
If we would have $-\infty < x < \infty$ and $-\infty < y < \infty$, I would try the double Fourier transform, but since it is given that x, y > 0, I have no idea what to do.
Edit: I thought, maybe I should use a Fourier sine transform in x, a Fourier cosine transform in y or both? But I don't see how to proceed.
The boundary condition is automatically satisfied if $u$ is defined on the entire plane as being odd in $x$ and even in $y$. So try first extending $f$ to the entire plane to have appropriate symmetries.