Timeline for What is the structure of the space of solutions of a non linear ODE?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 19, 2012 at 17:34 | comment | added | Qfwfq | I didn't know whether to accept R.Bryant's or F.Ziegler's very well written answer. The latter just seemed a bit more general, so I've clicked on it. | |
| Dec 19, 2012 at 17:30 | vote | accept | Qfwfq | ||
| Dec 8, 2012 at 23:29 | answer | added | Robert Bryant | timeline score: 8 | |
| Dec 8, 2012 at 22:50 | answer | added | Francois Ziegler | timeline score: 13 | |
| Dec 8, 2012 at 21:17 | answer | added | Alexandre Eremenko | timeline score: 9 | |
| Dec 8, 2012 at 20:55 | comment | added | Qfwfq | @Dean Yang: yes, that is an issue too; that's why I didn't specify in which functional space the solutions have to live ($\mathcal{C}^{\infty}(\mathbb{R})$ or $\mathcal{C}^{\infty}(t_0-\epsilon,t_0+\epsilon)$); should I have considered solutions as living in the space of germs of smooth functions at $t_0$ (which is still infinite dimensional)? ( @both commenters: I realize my question is perhaps a bit elementary for people who know differential equations). | |
| Dec 8, 2012 at 20:55 | comment | added | Qfwfq | @Robert Bryant: your comment comes very close to what I would have expected as an answer! Are there cases in which the manifold is compact? And in which it has nontrivial topology? | |
| Dec 8, 2012 at 20:30 | comment | added | Deane Yang | In addition to what Robert already said, there's also the issue of whether you care about the size of the interval on which a solution exists. For a linear ODE, the solution exists for all time (both positive and negative), but for a nonlinear ODE, solutions may exist for only a finite time and the size of the time interval might depend on the initial data. | |
| Dec 8, 2012 at 20:25 | comment | added | Robert Bryant | You need to put more assumptions on $F$ in order for this to have a meaningful answer. For example, you haven't ruled out that $F$ is identically $0$ (all functions $u$ satisfy the equation) or $1$ (no functions $u$ satisfy the equation). Usually, in the case of a higher order ODE for one unknown function, one assumes that the equation can be solved smoothly for the highest derivative. Then one locally has the structure of a smooth manifold of dimension $n$, with the 'initial values' $\bigl(u^{(j)}(t_0)\bigr)$ for $0\le j< n$ giving local coordinates. | |
| Dec 8, 2012 at 20:08 | history | asked | Qfwfq | CC BY-SA 3.0 |