↧
Answer by Funktorality for Nontrivial invariant transformations for heat...
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 space!
View ArticleAnswer by Iosif Pinelis for Nontrivial invariant transformations for heat...
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...
View ArticleNontrivial invariant transformations for heat equations
It is well known that if $u$ is a harmonic function on $\mathbb R^2$ then its Kelvin transform defined by$$ v(r,\theta) = u(\frac{1}{r},\theta)$$is also harmonic for $r>0$. Note that the Kelvin...
View Article
More Pages to Explore .....