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Consider the following PDE $$ \frac{\partial \Phi}{\partial t} - \frac{1}{2}y \frac{\partial \Phi}{\partial x} + \alpha \beta y^{3/2} \frac{\partial^2 \Phi}{\partial x \partial y} + \frac{1}{2} y \frac{\partial^2 \Phi}{\partial x^2} + \frac{1}{2} \alpha^2 y^2 \frac{\partial^2 \Phi}{\partial y^2} = 0 $$ What is a good substitution to solve this PDE ? I once used affine change of variable of type $$ \Phi(t,x,y) = \exp\left\{ A(t)x+B(t)y\right\} $$ and then deal with Ricatti PDE but it does not seem to help here.

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    Are you interested in a closed form solution?2012-09-25
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    Yes exactly, I do not care here about numerical approaches.2012-09-25
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    When $\beta=0$ , solve it by separation of variables. When $\beta\neq0$ , solve it by separation of variables + kernel method.2012-09-26
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    @doraemonpaul : any reference for the kernel method ? Thanks for your ideas.2012-09-26

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