Timeline for Heat conduction type equation in 4D
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| Dec 6, 2022 at 13:04 | comment | added | Fetchinson0234 | Also, in the earlier question about Green's functions, there is an analytic Green's function, $\phi(x) = e^{\xi x_4} / x^2$. This is analytic in $\xi$ but does not decay correctly for real $\xi$. For real $\xi$ there is another one with the correct decay. Similarly, I'd think there is the analytic heat conduction solution, $u(t,x) = e^{\xi t} e^{-x^2/4/t} / 4 / t^2$ but for real $\xi$ there should be another one, which leads to the aforementioned correct Green's function for real $\xi$. | |
| Dec 6, 2022 at 12:55 | history | edited | Fetchinson0234 | CC BY-SA 4.0 |
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| Dec 6, 2022 at 12:55 | comment | added | Fetchinson0234 | That may very well be, but then the statement is that some Green's functions (non-decaying ones) can be obtained from the heat kernel but some other Green's functions (decaying ones) can not? It may be, but I was not aware of such a subtlety. And what singles out the non-decaying Green's functions over the decaying Green's functions? So this entire state of affairs is a bit mysterious to me, but I might be overlooking something obvious. | |
| Dec 6, 2022 at 12:23 | comment | added | Carlo Beenakker | the issue is not whether $\phi=\int_0^\infty u\,dt$ satisfies the equation for the Green's function, which it does, but whether the function $\phi$ you obtain in this way decays when $x_4\rightarrow\pm\infty$; the integrand does decay, but there is no guarantee this decay is conserved upon integration. | |
| Dec 6, 2022 at 11:30 | comment | added | Fetchinson0234 | @Carlo Beenakker, I added some reasoning why I think it should be possible to reproduce the correct Green's function for real $\xi$. It's possible that I'm misunderstanding something of course. | |
| Dec 6, 2022 at 11:29 | history | edited | Fetchinson0234 | CC BY-SA 4.0 |
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| Dec 6, 2022 at 9:17 | answer | added | Carlo Beenakker | timeline score: 1 | |
| Dec 5, 2022 at 23:31 | history | edited | Fetchinson0234 | CC BY-SA 4.0 |
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| Dec 5, 2022 at 23:24 | history | asked | Fetchinson0234 | CC BY-SA 4.0 |