Partial differential equation problem

1. Jan 15, 2008

germana2006

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 the following partial differential equation

$$\partial_T \phi = \partial^2_X \phi (D_{||} + v^2 T^2 D_{\perp}) + D_{\perp} \partial^2 _Z \phi - 2 vT D_{\perp} \partial_Z \partial_X \phi - vT \partial_Z D_{\perp} \partial_X \phi + \partial_Z D_{\perp} \partial_X \phi$$

where v is the derivative of u. This partial differential equation is not easier to solve.

My idea is to apply the Fourier transformation for example to $$D_{||}\partial^2_X \phi$$ but then I become the convolution for the Fourier transformation for each one (D and $$\phi$$). Here it is my problem to apply the convolution in the partial differential equation.

I know that it is not very easy, but can someone help me.