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)

Then I mean the following are the same:

a. a distributional solution which has no equivalent solution to the distributional formulation of a classical equation

b. a solution of a distributional formulation which has no equivalent classical formulation

In other words, a solution is also a part of the formulation. So, the current definitions look OK for me. Do I overlook something? --Usagiop (talk) 00:11, 14 November 2022 (UTC)

Looks better. Here is another iteration of the same. In our context a differential equation can be formulated either classically or distributionally or both. For distributional formulation solutions can be either of classical type (involves ony distributions induced by classical functions), weak type (differential operators act on distributions induced by classical functions, but the free term is not necessarily so), or distributional type (both free and differentiated distributions dp not need to be induced by any classical function). Then the set of classical type of solutions $\subseteq$ the set of weak type of solutions $\subseteq$ the set of distributional type of solutions.--Julius (talk) 07:10, 14 November 2022 (UTC)

If you have a concrete problem, say in physic, you can formulate it classically or distributionally or anything else. A solution is then a differentiable function or distribution or something else, respectively. But if there exist multiple formulations for the same problem, then their solutions should be somehow the same as this theorem says. --Usagiop (talk) 00:17, 14 November 2022 (UTC)

Yes. The confusion here is that the differential equation has both classical and distributional formulations, and I wrote that the "distributional solution" has a certain form. But since everything is induced by classical functions, the solution can also be "weak" or "classical". However, it is strictly a solution of distributional formulation. So it surely cannot be a classical function. The domain is not real numbers, but test functions - a completely different mapping. But indeed it can be synonymous to a solution in some other formulation.--Julius (talk) 07:10, 14 November 2022 (UTC)

To get back on track, we have that weak solutions constitute a large variety of solutions, where distributional solutions occupy only a small part. Why do we need an additional page? Distributional notion in this case is narrower, so it should be fine, shouldn't it?--Julius (talk) 22:11, 14 November 2022 (UTC)

No idea. There are really various types of week formulations and solutions. We need to add more materials. I miss the classical Sobolev spaces in $\mathsf{Pr} \infty \mathsf{fWiki}$. Then fractional Sobolev spaces and Besov spaces and ... --Usagiop (talk) 22:42, 14 November 2022 (UTC)

Suggest you may want to find a source and make a start. --prime mover (talk) 23:54, 14 November 2022 (UTC)

I mean the complexity expands, exponentially, if we start adding new pages for all weak formulations. So I would suggest adding more materials preserving the current structure for now. --Usagiop (talk) 00:30, 15 November 2022 (UTC)