Timeline for Is this PDE solvable?
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 1, 2021 at 6:04 | comment | added | Laithy | Thank you Andrew, Terry, and Michael. :) I was trying to use Perron method since solutions to the PDE satisfy the maximum principle. But I was stuck. This is easier; it seems I get existence and uniqueness for the PDE. | |
| Feb 1, 2021 at 5:44 | vote | accept | Laithy | ||
| Feb 1, 2021 at 4:09 | comment | added | Michael Renardy | It should read $r^2\lambda''(r)+2r\lambda'(r)-b\lambda(r)+\lambda(1)=0$. This is an Euler equation. No Bessel functions needed. | |
| Jan 31, 2021 at 23:49 | comment | added | Terry Tao | Replace $\lambda(1)$ by an unspecified parameter $c$ to make the ODE local and then add the additional boundary condition $\lambda(1)=c$. Presumably the ODE you obtain can be solved exactly using Bessel or Hankel functions, either by hand or by using some standard symbolic computing package (Maple, Mathematica, SAGE, etc.). You may end up with some implicit equation for $c$ that then requires further analysis to solve but at least no further differential equations are involved. | |
| Jan 31, 2021 at 23:07 | comment | added | Laithy | you get an ODE that looks similar to the PDE: $\lambda''(r) + 2r \lambda'(r) - b\lambda(r) + \lambda(1)$ with boundary conditions $\lambda(1) + a \lambda'(1) = $known. (b is some constant depending on the eignevalue of the harmonics). How do I solve that? It also has a nonlocal component. | |
| Jan 31, 2021 at 6:35 | history | answered | Andrew | CC BY-SA 4.0 |