Or are there counterexamples? Especially some that are not quite artificial by e.g. taking the limit of a PDE series?
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asked Nov 5, 2012 at 12:11
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$\begingroup$ It's unclear what you're asking. If "formulating a PDE's analytical solution" means that you're exhibiting an explicit solution of the equation, then naturally that means that a solution exists...? $\endgroup$– hmakholm left over MonicaNov 5, 2012 at 12:15
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$\begingroup$ @HenningMakholm Yes that's my question - does one no longer have to prove the existence of a solution if one has already obtained it as an explicit expression, or is there some mathematical subtlety where this can fail? $\endgroup$– Tobias KienzlerNov 5, 2012 at 12:32
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$\begingroup$ If you have something explicitly in your hand, then of course whatever it is you have exists. (That the thing you have is actually a solution is of course something you need to prove somehow). $\endgroup$– hmakholm left over MonicaNov 5, 2012 at 15:22
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The short answer is, as noted by Henning, yes. If you have the analytical expression of a solution, then you have a solution. This is maybe the best scenario.
What you do not have is whether there are also other solutions to the problem. How can you be sure you have all possible solutions?
Often it is also difficult to see properties of solutions if the formula is complicated. But in most cases, even for simple ODEs, we are not able to write out the exact solution. It is more the lucky special case if you have one.
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$\begingroup$ Good point about the lack of completeness, although a mere prove of existence will of course also not reveal that... $\endgroup$ Nov 5, 2012 at 15:25