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(This is a repost of a question posed in StackExchange that didn't get any replies.)

Is anything known about the fundamental solution to the equation: $$\nabla^4 (Au) + \nabla^2 (Bu)+Cu=0$$ for constants $A$, $B$, and $C$ in 3D, $u=u(x,y,z)$.

For the equation: $$\nabla^4 u=0$$ The fundamental solution is $$u=r$$ where r is the distance from the singularity.

So, the question is how the additional terms modify the fundamental solution. References are most welcome.

It is surprising but interesting that information on this equation is difficult to find in literature.

The original post is here: https://math.stackexchange.com/questions/4581407/fundamental-solution-to-biharmonic-equation-with-second-order-and-zeroth-order-t

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    $\begingroup$ 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. $\endgroup$
    – Igor Khavkine
    Nov 24, 2022 at 1:05
  • $\begingroup$ 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 $\endgroup$
    – Talmsmen
    Nov 24, 2022 at 4:16
  • $\begingroup$ 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. $\endgroup$
    – Talmsmen
    Nov 24, 2022 at 4:16
  • $\begingroup$ 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 $C=1$). $\endgroup$
    – Jap88
    Nov 24, 2022 at 16:45
  • $\begingroup$ Correction: should be $-\nabla \cdot (R^2 \nabla u)+u=0$, also called screened Poisson's equation when written on this form. $\endgroup$
    – Jap88
    Nov 24, 2022 at 18:50

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