Talk:Partial differential equation

Notation and examples

Heat equation

In the section Heat equation, the constant k is usually referred to as thermal diffusivity.
Tianran Chen 03:56, 2004 Mar 1 (UTC)

Conceptual Definition

Arnold defines an ODE in his book as a system evolving in time having the following properties: (1) determinancy (2) finite dimensionality (3) smoothness

where by determinancy he means that the initial data of the problem completely specifies the future and the past of the system. By a finite dim. system he means one whose phase space can be localy parametrized by finite many real numbers. The smoothness property means that the system has a smooth phase space (ie. the phase space is a differentiable manifold, eg. The plane, the real line) and smooth evolution function.

What i would like to know is if there exists such a nice conceptual definition of a PDE.

Assessment comment

The comment(s) below were originally left at Talk:Partial differential equation/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The front-line article on PDEs should cover vector-valued PDEs (aka, systems, which are mentioned briefly). Fundamental concepts such as underdetermined, determined, and overdetermined PDEs need to be discussed more systematically. Geometry guy 22:16, 9 June 2007 (UTC)

Last edited at 22:16, 9 June 2007 (UTC). Substituted at 02:27, 5 May 2016 (UTC)

Can someone please add a section on the unified transform (Fokas) method

We need a section on the unified transform method, first published by Athanassios Fokas. The number of mathematicians researching this area is large, and so is the number of applications. Perhaps someone who can explain it properly can help us out? Yoodlepip (talk) 20:11, 1 January 2017 (UTC)

This method might be deserving of a mention here, but if it is as important is you are implying, it should probably have its own article. Direwolf202 (talk) 14:14, 22 April 2019 (UTC)

Classification section

It doesn't seem to me that the classification of PDEs needs so much coverage compared to other things. Should we create a separate article for the classification, going into full detail and implications, while adding a word on terminology like "order" and say simply that PDEs can be classified according to their behaviour, with a link to that separate article? Direwolf202 (talk) 14:20, 22 April 2019 (UTC)

Introduction needs rewriting

The first paragraph of the Introduction is as follows:

"Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. For example, the position of a rigid body is specified by six parameters, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer."

This says nothing at all about why partial differential equations are more like the configuration of the fluid rather than like the rigid body.

It is as if the writer forgot what they were going to write.

So this definitely needs clarification! 50.205.142.50 (talk) 13:10, 27 March 2020 (UTC)