Timeline for Second order linear ODE with mixed boundary condition
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 8, 2011 at 15:40 | vote | accept | ProbLe | ||
| May 8, 2011 at 9:39 | comment | added | ProbLe | I would like to thank all of you for your thinking on my some what stupid question. For the case when $a(t)$ is a constant function, it is easy to disscus and, as Deane said to me, for the general case the theory of Sturm-Liouville can help. Well, if you do think this question is not suitable for MO, we could delete it. Thanks! | |
| May 7, 2011 at 18:26 | answer | added | Deane Yang | timeline score: 1 | |
| May 7, 2011 at 8:59 | comment | added | Igor Khavkine | Each of the above cases can be explored with $a$ constant, as per Deane's suggestion. For arbitrary $u$ and $a(t)$, it is a non-trivial problem to figure out which of the cases you are in, and has to be tackled separately for each case (using for instance exact solutions, qualitative estimates, or numerics). BTW, a nice modern reference is Zettl, Sturm-Liouville Theory. amazon.com/dp/0821839055 | |
| May 7, 2011 at 8:56 | comment | added | Igor Khavkine | Generically, both existence and uniqueness will be satisfied (solution space is 2-dimensional, there are two constraints on it). However, if $u\ne 0$, a solution may fail to exist at all if $f(1)\implies df/dt(0)=0$. On the other hand, if $u=0$, existence is trivial ($f(t)=0$), but uniqueness is not guaranteed. | |
| May 7, 2011 at 1:19 | comment | added | Thierry Zell | I don't think this is quite MO-level, though I'm sure there could be a very interesting discussion to be had around your question. My advice would be to assume that the solution is not unique in general, because boundary conditions seldom lead to uniqueness. So I would look for a counterexample. A much trickier question will be: are there easy conditions to check for which we do have uniqueness. This might be what Deane has in mind, but I can't be sure. | |
| May 6, 2011 at 21:15 | comment | added | Deane Yang | Before someone tells you some answers, it is rather instructive to work out examples yourself. If you let $a$ be a constant (the sign matters!), you can find the general solution to the ODE (without the boundary conditions) and figure out when the boundary conditions can be met or not. After you've done that, look up "Sturm-Liouville theory" for self-adjoint second-order ODE's. | |
| May 6, 2011 at 21:04 | history | asked | ProbLe | CC BY-SA 3.0 |