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?