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A nonlinear PDE that has been bugging me for months, hoping someone has an idea, it looks so simple! Consider the following, for $u(x,t)$

$D \displaystyle\frac{\partial u}{\partial t} = u^2 \displaystyle\frac{\partial^2 u}{\partial x^2}$

subject to boundary conditions

$u(1,t) = 1$,

$\displaystyle\frac{\partial u}{\partial x} (1,t)= t$,

and initial condition

$u(x,0) =1$.

The standard similarity solution $u(x,t) = \displaystyle t^{\alpha}F\left(\frac{x}{ t^{\beta}}\right) $ fails due to the second boundary condition giving the same equation for $\alpha$ and $\beta$ as the PDE.

Any thoughts on linearising this, or other methods to obtain a solution would be much appreciated!

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    Also, I had a mistake in the B/C, thankyou.2012-12-04

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