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)