I'm trying to solve the following PDE (derived from convective heat transfer in a fluid flow between two parallel plates) analytically and I'm not sure what path to take. I don't think separation of variables or using Laplace transforms will work and there doesn't seem to be an obvious (to me) direct way to solve.
$\epsilon Y(1-Y)\frac{\partial T}{\partial X} = \gamma (1-4Y+4Y^2) + \kappa \frac{\partial T^2}{\partial Y^2}$
subject to:
$T(0,Y) = 1-4Y(1-Y)$
$T(X,1) = 1$
$T(X,0) = 1$
In the first step, the PDE is transformed to a canonical form :
In the second step, a family of particular solutions is derived (involving Parabolic Cylinder functions) :
In the third step, the general solution is expressed on the form of series of the above solutions or on integral form :
The last step would consist in optimizing the coefficients $C_{\lambda}$ or the function $C(\lambda)$ to fit with the boundary conditions. This is the more problematic part of the job.
Note: Some particular parabolic cylinder functions (for particular values of $\lambda$) are related to Hermite polynomials. This gives hope to a simpler form of solution involving Hermite polynomials.