New answers tagged parabolic-pde
6
votes
Nontrivial invariant transformations for heat equations
Yes, how about the Appell transform, see here.
$$v(t,x)\mapsto\Gamma(t,x)v(-\frac1t,\frac xt)$$
where $\Gamma$ is the heat kernel.
Truly nontrivial if you ask me. Of course, this inverts time and not ...
3
votes
Accepted
Nontrivial invariant transformations for heat equations
At least when $n=1$, there are no nontrivial transformations of this kind.
Indeed, suppose that $v(t,x)=u(\tau(t,x),\xi(t,x))$, where $u_t=u_{xx}$. Then $u_{tx}=u_{xxx}$ and $u_{tt}=u_{xxxx}$, so that ...
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