Talk:Method of characteristics

Rarefaction

It's not clear (to me) how the description of the solution in a 'rarefaction' as only existing in a weak sense tallies with the normal use of terms in fluid flow. The solution in such area (IIRC) would often be an isentropic self-similar expansion, ie a physically acceptable solution, which is quite well determined. Do mathematicians need to take a paragraph off to educate physicists here? Linuxlad 17:54, 26 June 2006 (UTC)

What Happened to the Plot?

The article says, "A solution is shown in the figure below as a surface plot and a contour plot". I've checked on two different computers...seems like the plot is gone. Can someone put it back up?

and another thing ... the example is really specific and does not show throughly the method and again can someone add the plot?

Example

The example is a linear example. I think one fully non-linear example would be great to add to this article, since the compuations are quite different MATThematical (talk) 20:12, 18 October 2009 (UTC)

Higher-order equations

The article claims that this method works for any higher-order hyperbolic equation, but I really can't see how it can work if different power derivatives in same variable (e.g., $\partial _{x}$ and $\partial _{x}^{3}$) appear. Njerseyguy (talk) 06:27, 13 May 2010 (UTC)

OK, I think I found the point of confusion. I think in this article, "Hyperbolic" refers to 2nd-order equations with $b^{2}-4ac\geq 0$, whereas the article on Hyperbolic partial differential equation, which the word "hyperbolic" links too, is much more general (including nth-order equations). Can anyone confirm? Njerseyguy (talk) 17:27, 13 May 2010 (UTC)

It is only the leading symbol of the operator that determines the characteristics, so in the example where $\partial _{x}$ and $\partial _{x}^{3}$ both appear, it is only the highest order terms that contribute to the characteristics, the idea being that the Cauchy data propagates along the characteristics, and solvability of the Cauchy problem depends only on the highest order terms (which is more or less obvious if you try a formal power series solution). That said, it is perhaps overly optimistic to say that the "method works" for any higher-order hyperbolic equation. Certainly the method of characteristics is important in those cases (in so far as there is a well-defined "method" to speak of), precisely because the initial Cauchy data propagate along characteristics. But one will never get a complete answer from the method in those cases, except in very simple examples like the one-dimensional wave equation. Sławomir Biały (talk) 14:25, 14 May 2010 (UTC)

related ideas

I think it would be good if this article compared method of characteristics with method of lines, and also mentioned some of the applications/extensions (e.g. modified method of characteristics as in hopmoc)? Cesiumfrog (talk) 05:32, 22 July 2015 (UTC)

Missing definitions in subsection "Linear and quasilinear cases"

What are $\mathbf {X} (s)$ and $U(s)$? I can suppose $\mathbf {X} (s)=(x_{1}(s),\ldots ,x_{n}(s)),$ but $U(s)$? It is perhaps $u(s)$? — Preceding unsigned comment added by Enzotib (talk • contribs) 10:14, 11 October 2020 (UTC)

Historical background

Does anybody know where the method originates? Who started using it and what for?

M. B Abbott - An introduction to the method of characteristics -Thames & Hudson (1966), is saying that initial study of wave propagation was done by D'Alembert and Goursat in the 1850s, and Riemann was the first to apply the method of characteristics to gas dynamics, but maybe someone is more informed than me?

Lockywolf (talk) 03:15, 27 May 2021 (UTC)

I'm also not sure of the history, but I agree that something along this lines would be really interesting. There must be a link to Lagrange and Charpit, although as usual I suspect that they didn't originate the method which now bears their names! :) plw (talk) 01:21, 27 May 2022 (UTC)

Try Lagrange and fr:Paul Charpit. The French article states clearly that multiple authors give Charpit credit for the idea. Charpit did not provide rigorous proofs. His diary/manuscript circulated among several famous mathematicians, but then was lost until 1928. There are two known copies; the manuscript was never published. 67.198.37.16 (talk) 04:25, 24 March 2024 (UTC)