Hey I am trying to solve following PDE:
$\frac{\partial \Psi}{\partial t}+Ax_2\frac{\partial \Psi}{\partial x_1}=\nu (\frac{\partial^2 \Psi}{\partial x_1^2}+\frac{\partial^2 \Psi}{\partial x_2^2}+\frac{\partial^2 \Psi}{\partial x_3^2})$
I tried using the seperation of variables as $Psi(x_1,x_2,x_3,t)=F_1(x_1)F_2(x_2)F_3(x_3)T(t)$
However I do find it diffult to solve this, since in the PDE it seems the dependency between $x_1,x_2$ is very linked... MAPLE also could not really help.
Any advice or someone who could provide me with an approach to solve this PDE?
I would appreciate any kind of help.
Using separation of variables, and by assuming $F_1(x_1)=e^{r_1x_1}$, $F_3(x_3)=e^{r_3x_3}$, $T(t)=e^{r_4t}$, you're left an ODE describing $F_2(x_2)$, \begin{align} \frac{\mathrm d^2F_2}{\mathrm dx_2^2}+\left(r_1^2+r_3^2-\frac{r_4+Ar_1y}{\nu}\right)F_2=0, \end{align} whose solution is some sort of Airy function. https://www.wolframalpha.com/input/?i=f%27%27%2B%28a%2Bby%29f%28y%29%3D0