Timeline for Fundamental Solution to Biharmonic Equation in 3D
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 24, 2022 at 18:50 | comment | added | Jap88 | Correction: should be $-\nabla \cdot (R^2 \nabla u)+u=0$, also called screened Poisson's equation when written on this form. | |
| Nov 24, 2022 at 16:45 | comment | added | Jap88 | Thanks both . The question comes from an applied math application where the equation is used to filter and interpolate data in 3D space. Helmholtz equation gives a natural length scale $R$ through $\nabla \cdot (R^2 \nabla u)+u=0$ and $u \approx \frac{e^{-r/R}}{r}$. The question then is which natural length scales (if any) that $A$, $B$, and $C$ correspond to. (One can of course set $C=1$). | |
| Nov 24, 2022 at 4:16 | comment | added | Talmsmen | ned to think that the Fourier method would be most ideal as it is the de facto choice among physicists studying extension theory in General relativity. | |
| Nov 24, 2022 at 4:16 | comment | added | Talmsmen | There is also the Michell solution for the Biharmonic equation. (en.wikipedia.org/wiki/Michell_solution) It really just boils down to making a variable substitution $r \mapsto e^{t}$ and factoring a quartic polynomial. If I recall correctly, IIT has lecture notes available for the derivation. Unfortunately, $B\Delta u$ and $Cu$ will ruin the necessary symmetry for the change of variables. That being said, you could still produce a recurrence relationship that could probably be solved numerically. Of what theoretical value such a formula would be is outside of my knowledge. I am incli | |
| Nov 24, 2022 at 1:05 | comment | added | Igor Khavkine | Are you not happy with the Fourier integral representation $G(r) = \int \frac{d^3p}{(2\pi)^3} \frac{e^{ip\cdot r}}{A|p|^4-B|p|^2+C}$? This approach works for any constant coefficient equation. | |
| Nov 23, 2022 at 22:31 | history | asked | Jap88 | CC BY-SA 4.0 |