I consider the typical convection-diffusion equation taking into account the diffusion coefficient as a tensor like:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]D= \left( \begin{array}{cc} D_{||}(y) & 0 \\ 0 & D_{\perp}(y) \end{array} \right) [/tex]

then, the equation will be:

[tex]\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)][/tex]

In the case of the [tex]D = cte [/tex], I can solve this equation doing a change of variable and then applying the Fourier transformation. But when [tex]D[/tex] is a tensor, I become the following partial differential equation

[tex]\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 [/tex]

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 [tex]D_{||}\partial^2_X \phi[/tex] but then I become the convolution for the Fourier transformation for each one (D and [tex]\phi[/tex]). 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.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Partial differential equation problem

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads for Partial differential equation | Date |
---|---|

A How to simplify the solution of the following linear homogeneous ODE? | Feb 18, 2018 |

I Substitution in partial differential equation | Jul 8, 2017 |

A A system of partial differential equations with complex vari | Jul 6, 2017 |

**Physics Forums - The Fusion of Science and Community**