## differential equation problem

I consider the typical convection-diffusion equation taking into account the diffusion coefficient as a tensor like:

$D= \left( \begin{array}{cc} D_{||}(y) & 0 \\ 0 & D_{\perp}(y) \end{array} \right)$

then, the equation will be:

$\frac{\partial \phi(x,y,t)}{\partial t} + \vec{u} \cdot \vec{\nabla}\phi(x,y,t) = \vec{\nabla}[D \cdot \vec{\nabla} \phi(x,y,\tau)]$

In the case of the $D = cte$, I can solve this equation doing a change of variable and then applying the Fourier transformation. But when $D$ is a tensor, I become a partial differential equation that I cannot solve. I make the change of variable but I cannot applied the Fourier transformation because $D$ depends on $y$.

Could some one give me other idea to solve it?