Talk:Distributional Solution to y' - k y = 0

Classical vs weak vs distributional solution to a differential equation
The source I am using is not very clear on how weak and distributional solutions differ from each other. However, I have managed to pull the following classification from a different source.

Let $\Omega \subset \R^n$.

Let $P$ be a differential operator of order $m$.

Consider the equation $Pu = f$.


 * $u$ is a classical solution if $u \in \map {\CC^m} \Omega$, and $f$ is also some ordinary function.


 * $u$ is a weak solution if $u$ is locally integrable in $\Omega$, but still an ordinary function, and $f$ is a distribution, either induced by a nice function or something like Dirac delta or its derivatives.


 * $u$ is a distributional solution if both $u$ and $f$ are distributions, either usual functions or something like Dirac deltas or their derivatives.

Now a classical solution can be weak or distributional, but it does not work the other way around. I guess we will use the least general case for each case, with the distributional solution being the most general one. This classification does not say anything about nonlinear equations, though.--Julius (talk) 21:06, 4 November 2021 (UTC)


 * Probably best to keep the explain template and redlink in place until we have some solidity here. Might be worth raising the question on MathStackExchange (citing the reference and giving its exposition), we might get somewhere with that. This is a bit far over my head at this stage. I seem to have Willmore's "Introduction to Differential Geometry" (1959) on my shelf but never cracked it open till now. It doesn't really go into distributions, and the first time it mentions them is on page 243, and so there's a lot of study before I get remotely anywhere near there. --prime mover (talk) 22:23, 4 November 2021 (UTC)


 * Sadly, distributions in differential geometry mean something completely different, so your book would not help. Instead, we need a book on generalised functions or not-so-introductory text about Fourier transform. Anyway, I will keep this in mind and adjust these red links accordingly.--Julius (talk) 06:18, 5 November 2021 (UTC)

Is the solution distributional or weak?
There are at least two ways to think of this problem. Firstly, we have classical and distributional formulation of the differential equation. Then a solution is classical or distributional if it solves one of these formulations. Sometimes there is a direct correspondence between both formulations. Secondly, for a distributional formulation we can have classical, weak, and distributional type of solutions. Classical type is the one where we can replace distributions by associated classical functions. The weak type is the one where RHS is purely distributional. The distributional type is the one where both sides involve purely distributional functions. So I guess we have an issue of degeneracy of terminology here. I think by distributional I had in mind the first classification, i.e. it solves the distributional formulation of the differential equation. According to the second classification this should be classical because no pure distributions are involved. But I need to look more into the terminology.--Julius (talk) 20:47, 13 November 2022 (UTC)

To sum, we should distinguish a solution to classical/distributional formulation of differential equation from a distributional solution which only applies to distributional formulations.--Julius (talk) 21:04, 13 November 2022 (UTC)


 * Not sure but I think the following are the same:
 * 1. (what you call) a distributional solution of a classical formulation of a classical differential equation
 * 2. a solution of a distributional formulation of a classical differential equation
 * So there seems nothing to distinguish. --Usagiop (talk) 21:49, 13 November 2022 (UTC)


 * Indeed these two things are the same. However, by "a distributional solution which only applies to distributional formulations" I mean a much larger family solutions involving Dirac delta and its relatives. Now there may be a distributional solution but no equivalent solution to the distributional formulation of a classical equation because Dirac delta cannot be made classical.--Julius (talk) 22:08, 13 November 2022 (UTC)