Timeline for Question on PDE
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 26, 2011 at 2:46 | comment | added | Turbo | "Getting the PDE to stop diffusing without introducing an artificial stopping time would be more subtle." Actually I think I am interested in exactly this "automation" of the stopping times so that the description of the PDEs themselves would capture the closest neighbors. Such a description could provide an analytical gadget to some euclidean graph problems | |
| Apr 26, 2011 at 2:27 | comment | added | Aaron Hoffman | (continued) ... and then call a collision any time that $|A_i - A_j| \ge r_i + r_j$. Getting the PDE to stop diffusing without introducing an artificial stopping time would be more subtle. | |
| Apr 26, 2011 at 2:24 | comment | added | Aaron Hoffman | Caveat: Maybe I'm not understanding your question. I am imagining $u_t = D\Delta u$ where $u(x,t) \in \R^m$ and $x \in \R^n$ and $D$ is a diagonal matrix with entries $k_i$. This is just a collection of uncoupled heat equations in $\R^n$ whose fundamental solution is just a vector of Gaussians. It sounds like the solution that you are interested in corresponds to $u_i$ a Gaussian centered at $A_i$. The stopping time is introduced artificially by picking levels $\epsilon_i$ and taking logarithms of the Gaussians to solve for the radius of the spherical level set as a function of time ... | |
| Apr 26, 2011 at 1:55 | comment | added | Turbo | I am wondering what the system would look like for more than $3$ points in $n \ge 3$ dimensions. | |
| Apr 26, 2011 at 1:54 | comment | added | Turbo | Hi Aaron: Thank you for the answer. Can you write explicitly the system of PDEs that will start diffusing and stop at the closest neighbor's level set? | |
| Apr 25, 2011 at 11:06 | history | answered | Aaron Hoffman | CC BY-SA 3.0 |