$$ \frac{\partial u}{\partial t} = t\nabla^2 u \longrightarrow u(x,0) = \delta(x) $$
Hi all, I was wondering what ways are possible to solve this. Using a scaling argument where $$u(\lambda x,\lambda^k t) = v(y,s)$$, but all I was able to obtain from this is the relation $$v_t = \lambda^{2-2k}\nabla^2 v $$ in the new coordinate system. I thought transforming this equation to be bounded on a disk would also work, in which I let $$r \longrightarrow \lambda r$$ & $$ t \longrightarrow \lambda t^2$$
From all this, I think you can say that $$ \lambda ~ t^{-1}$$ and plug in $$u(x,t) -> u(\frac{x}{t^2},1) = \Phi (r)$$ but I am not sure how this gets to a solution, or can be used towards the B.C. Should I rewrite the PDE in terms of $$ \Phi(r)$$so I can solve it?
Does anyone have any tips/help on how to get this into a form that I can integrate using Fourier transform? I'd appreciate any help.