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

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

then, the equation will be:

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

In the case of the $\displaystyle D = cte $, I can solve this equation doing a change of variable and then applying the Fourier transformation. But when $\displaystyle D$ 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 $\displaystyle D$ depends on $\displaystyle y$.

Could some one give me other idea to solve it?