Talk:Duhamel's principle

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The description of Duhamel's principle as "the solution to the inhomogeneous wave equation" is terribly inaccurate. Duhamel's principle is used to solve the inhomogeneous wave equation, the inhomogeneous heat equation, and even the inhomogeneous transport equation. It is the idea that these problems can be solved by integrating solutions to homogeneous problems in time. Evans PDE book is a decent reference. -- Ivan Blank, Assistant Professor of Mathematics at Kansas State University (My field of research is a subfield of PDEs.) —Preceding unsigned comment added by 70.179.160.248 (talk) 08:32, 8 December 2008 (UTC)

Moreover, the solution for inhomogeneous wave equation from this article is incorrect. The correct result(for PDE) will be the sum of solution for homogeneous equasion, given by D'Alambert formula, and Duhamel's integral from an article. 85.141.157.237 (talk) 18:40, 6 January 2010 (UTC)

The Duhamel idea can be used for more general spatial operators, not just constant coefficient ones. The constant coefficient setting just makes the proof easy in the space of tempered distributions (Fourier transform). It would be nice to see some more general cases discussed in the article. —Preceding unsigned comment added by 169.231.110.153 (talk) 17:35, 12 July 2010 (UTC)

The solution to the wave equation

This section is terribly confusing by not defining what $f(\xi ,s)$ is.

Moreover the lead only mentions PDEs first order in t, but the wave equation is second order in t. It is not clear how Duhamel's principle applies here. — Preceding unsigned comment added by 18.111.126.5 (talk) 16:38, 26 April 2014 (UTC)

I have no wish to defend the section as a great example of mathematical writing, but only to clarify some of these points. First, the section under discussion begins

Given the inhomogeneous wave equation:

$u_{tt}-c^{2}u_{xx}=f(x,t)\,$

The function f is the forcing term. Secondly, it is mentioned earlier in the article that Duhamel's principle also applies to linear systems of evolution equations, and therefore also to higher order evolution equations, since these can be described as an equivalent system. Sławomir Biały (talk) 17:00, 26 April 2014 (UTC)

Assessment comment

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Geometry guy 22:31, 10 June 2007 (UTC) 85.141.157.237 (talk) 18:41, 6 January 2010 (UTC) In response to whoever asked for more general statements: Duhamel's principle can be explained in a fairly general setting with the use of semigroup ideas... The book titled Introduction to PDE by Renardy & Rogers (Chapter 11, I think) is not a bad reference. In fact, the propagators are nothing else than exponentials of dissipative operators. Once the existence of a semigroup of propagators for u' = Au is ascertained, Duhamel's principle follows very easily. —Preceding unsigned comment added by 68.6.100.220 (talk) 00:09, 26 October 2010 (UTC)

Last edited at 00:10, 26 October 2010 (UTC). Substituted at 02:02, 5 May 2016 (UTC)

Constant-coefficient linear ODE

I wish there were more details about the statement that

$P(\partial _{t})H=\delta$

in the sense of distributions.

First of all is G a smooth function? so that H is the product of a smooth function times Heaviside? If that's the case then I would use Leibniz' formula for each monomial $a_{k}(\partial _{t})^{k}H$ but I'm not sure what it gives, Dirac's delta and its derivatives will appear as derivatives of the Heaviside. For $t>0$ the equation vanishes indeed, but at $t=0$ is it obvious that we only have $\delta$ and not a sum of derivatives of it multiplied by derivative of G... Ah, .... ok, I get it, the initial conditions on G allow to conclude!! — Preceding unsigned comment added by Noix07 (talk • contribs) 15:49, 3 September 2020 (UTC)

In fact I think there is a factor m missingNoix07 (talk) 16:04, 3 September 2020 (UTC)