Talk:Partial differential equation

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Mathematics rating:	B Class	Top Priority	Field: Analysis
One of the 500 most frequently viewed mathematics articles.
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. 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)

[edit] Bad Mathematics

This article is flawed from the onset. I tried to give it some rigour, but just gave up. For example: If u_x = 0 then u(x,y) is independant of x, and so u(x,y) = f(y) "where f is an arbitrary (differentiable) function of y..." Well, try u(x,y) = | y | . Clearly u_x = 0 for all x, but | y | is discontinuous at y = 0, and hence so too is .

But u(x,y) = | y | IS continuous everywhere. Saran T. (talk) 12:24, 21 April 2008 (UTC)

I think the real question may be: why does the article require a differentiable function? The above comments give a solution that is where f(y) is not differentiable.

Thenub314 (talk) 01:06, 22 April 2008 (UTC)

That requirement was added by an IP editor. I don't see any reason for it, so I removed it. -- Jitse Niesen (talk) 10:14, 22 April 2008 (UTC)

[edit] Notation and examples

[edit] 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)

[edit] Methods to solve PDEs

Separation of variable is a frequently used method of solving linear partial differential equation. I think we should have some description here. If no one disagree, I will put them in this section.
Tianran Chen 03:56, 2004 Mar 1 (UTC)

Functional Integral Could you solve a PDE (linear or not) using the Fnctional integral formalism? , in the article called functional integral they say these kind of integrals are used when solving PDE's or simply the difussion equation, Heat equation and Schröedinguer equation can be solved this way.--Karl-H 10:02, 29 January 2007 (UTC)

[edit] Classification

I don't understand the section on classification into hyperbolic, etc. Where is this stuff supposed to be coming from? And is this to imply that linear PDEs of order greater than 2 have a classification? The only classification method I've seen is B² − 2AC with B the coeffecient of u_xy and so on. --ub3rm4th 21:23, 23 Feb 2005 (UTC)

It's been added in. I may have made a mistake; I'm rather sleep-deprived and I haven't studied PDE since last spring. I'd be grateful if someone would edit what I've added and made the analogies I've skeletoned out more clear. --Eienmaru 18:46, 13 August 2005 (UTC)

This classification only seems to work for partial differential equations of two variables. What about the classification of pde's of more than two variables? Also, I have read (at http://www.cs.cornell.edu/Courses/cs667/2005sp/notes/13guerra.pdf) that this classification "is based on the equation’s curve of information propagation". This suggests to me that the classification into hyperbolic, parabolic and elliptic is related to the time variable/time derivative of the PDE, and yet the time variable is not mentioned in this classification. Does anybody have any comments/clarifications on this? Daisy Horin.

[edit] Discriminant

Hi, I am not very familiar with partial differential equations, but I do recognise some stuff from conics. I don't understand how you get to the formule b^2-2*a*c. Shouldn't that be b^2-a*c? I'm aware of your reasons to take 2*b at a point as a coefficient, and I am going along with that so that can't be the reason (if you didn't wouldn't it be something like b^2-4*a*c or so?

This site also says, within your convention, it is b^2-a*c http://csep1.phy.ornl.gov/pde/node3.html

I hope what I said makes some sense,

thanks

Absolutely. Should be b^2-a*c. I doublecheck in Courant's book. Thank you for the correction.

--GS 15:28, 1 September 2005 (UTC)

I changed the constants to make the discriminant look more like the familiar discriminant which appears in the quadratic formula. –Matt 07:42, 3 November 2005 (UTC)

[edit] First order system example removed

The matrix associated with the system

u_t + 2v_x = 0

v_t − u_x = 0

has coefficients,

The eigenvectors are (0,1) and (1,0) with eigenvalues 2 and -1. Thus, the system is hyperbolic.

Some problems with this:

1. The conclusion is wrong. Eliminating v from the system we find that u satisfies u_xx + 2u_yy = 0, meaning the system is elliptic, not hyperbolic.
2. It is not at all clear how the matrix is related to the coefficients of the system.
3. The concept being conveyed here is incorrect in the first place. Classification of first order systems is NOT a simple carrying over of the scheme for second order equations.

In the simplest case, that dealt with here, a homogeneous, 2 by 2, first order linear system for unknown functions u(x,y) and v(x,y), there are in general eight coefficients:

A₁₁u_x + A₁₂u_y + B₁₁v_x + B₁₂v_y = 0

A₂₁u_x + A₂₂u_y + B₂₁v_x + B₂₂v_y = 0

In general we have to deal with two matrices, A and B and, unlike the case for a single second-order equation, one cannot assume that the matrices are symmetric.

The whole topic of classification of first-order systems is worth an article in itself. I don't know all the details, but I believe the analysis hinges on the (possibly complex) roots of the equation det(A − λB) = 0. Brian Tvedt 02:54, 23 November 2005 (UTC)

[edit] 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.

[edit] Bad Mathematics?

I thank the creator and contributors of and to this section respectively. It is a daunting task to give a rigorous, concise and lucid exposition on partial differential equations (PDEs) when it is clear from the references alone that a comprehensive treatment may be beyond the capacity of an encyclopedia (written or electronic). That being said, I might suggest adding a few words about the general theory only being in the small.

Mentioned in another part of the Wikipedia is the concept of weak solutions. Weak solutions play an important role in the theory of PDEs. --Jss214322 (talk) 00:54, 2 February 2008 (UTC)