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I have a partial differential equation of the form $c_t = a_0 c_{xx} + a_1 c_{yy} + \left( a_2 y^2 + a_3 \right) c_x$ (where subscripts represent partial derivatives, $t$ is time, and $x$ and $y$ are spatial). I don't think it's possible to do separation of variables because of the product of $y^2$ and $c_x$. I don't have much experience with solving PDEs, though; unfortunately, I ran out of ideas when I didn't find this on the "List of nonlinear partial differential equations" Wikipedia page.

Is there any hope for solving this analytically? I have a numerical result to work with, but I'd like to have an analytical result where possible for plots and such. Is it possible to solve this for restricted conditions, such as where $a_2 y^2 + a_3 = 0$?

Thank you!

skypilot27
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    With the general coefficients $a_i$, there's not a whole lot we can do. If you have more information on these coefficients you could use a coordinate transformation to transform the principal part of the PDE into its canonical form, but the presence of the first order derivatives does complicate matters somewhat. EDIT: Also this PDE is actually linear, so searching the nonlinear PDE page won't bring you any matches. – K.defaoite Jan 22 '21 at 02:40
  • @K.defaoite Thank you. I know that $a_1=1$ in the particular case I'm looking at, but the rest are still symbolic. – skypilot27 Jan 22 '21 at 02:44

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