Talk:Partial differential equation/Archive 1

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)

First order system example removed

 * The matrix associated with the system


 * $$u_t+2v_x=0$$


 * $$v_t-u_x=0$$


 * has coefficients,


 * $$\begin{bmatrix}

2 & 0 \\ 0 & -1\end{bmatrix}$$


 * 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 $$u_{xx} + 2 u_{yy} = 0$$, 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:


 * $$ A_{11} u_x + A_{12} u_y + B_{11} v_x + B_{12} v_y = 0 $$


 * $$ A_{21} u_x + A_{22} u_y + B_{21} v_x + B_{22} v_y = 0 $$

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 $$det(A-\lambda B)=0$$. Brian Tvedt 02:54, 23 November 2005 (UTC)

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)

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 $$\partial |y| / \partial y$$.


 * 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)

Classification
I don't understand the section on classification into hyperbolic, etc. Where is this $$\!\, v_t-v_x$$ 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 $$B^2-2AC$$ 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.

In my opinion, the line on the classification of parabolic 2nd order pde's in n variables should read “2. Parabolic : At least one eigenvalue is zero.“, since it is defined as having the matrix with detA = 0. An example: 3 u_x1x1 + u_x2x2 + 4 u_x2x3 + 4 u_x3x3 + u_x4x4 + 6 u_x4x5 + 9 u+x5x5 = 0. 193.247.250.59 (talk) 21:37, 14 June 2010 (UTC)

Why dont we give a broad veiw of what partial differential equations really are
Hi, why dont we give a broad veiw of what partial differential equations really are before going into the technical stuff. I know its maths, and we all want to see rigour and the like, but for someone just wanting to get oriented in the matter this is not such a good place to come to. A bigger (maths lite) introduction to the matter is needed maybe. Plus, historical context would be good, important people and cases, etc. Thanks —Preceding unsigned comment added by 203.206.255.53 (talk) 15:05, 20 July 2010 (UTC)
 * I agree, though I'm in no position to offer specifics at the moment. I just posted a comment on the talkpage at Fourier series, the article which led me here. Much of that feedback applies to this piece, as well. - PrBeacon (talk) 07:51, 6 October 2010 (UTC)

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)

 Eigenfunction Expansion : I don't see a section for this method. Is anyone else familiar with it? It may be important b/c it transforms inhomogeneous PDEs & B.C.s into homogeneous ones, so separation of variables can be applied. For problems w/ homogeneous B.C.s, sep. of vars. really can't be used. So I think it could be important to include a section for e-function expansion method. —Preceding unsigned comment added by 75.173.40.108 (talk) 22:05, 21 May 2011 (UTC)

Spectral Methods
Is another important method to solve PDEs (numerically). --Jorgecarleitao (talk) 08:46, 2 November 2011 (UTC)

Inline citations missing?
There are 19 references for the article. There does not appear to be any inline citations? --Михал Орела 14:14, 25 January 2012 (UTC) — Preceding unsigned comment added by MihalOrela (talk • contribs)

Systems of first-order equations and characteristic surfaces
Perhaps I'm too ignorant on the subject but it seems to me that the coefficient matrices Aν need not be m by m matrices. Can't the system comprise other than m equations? In this more general case Aν could then have p rows (as many as the vector B, I guess). 129.240.215.211 (talk) 15:51, 30 May 2015 (UTC)

Intro paragraph question
The very beginning of the intro discusses the equations governing dynamics of solid bodies and those governing fluid dynamics. Then says this distinctions make PDEs much more difficult to solve. This is a non-sequitur, first of all, which is the PDE, I assume fluid dynamics use partial differential equations. Second, it does not make sense that the specific example makes PDE in general more difficult to work with, unless you state that the differences in this example generalize to all types of PDEs versus regular differential equations, and I don't think that is the case. The example only clouds the entire paragraph. The fact that PDE involve equations that change with the value of other continuous variables is not at all illustrated by the examples, not without additional explication. I am not the one to fix this, however. — Preceding unsigned comment added by 50.72.161.240 (talk) 00:18, 15 September 2015 (UTC)

The sentence,
 * "Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic"

seems suspect to me. Mathematical descriptions usually involve idealizations, not underlying dynamics. So maybe the sentence should replace dynamic with theory? Rhetth (talk) 22:56, 19 April 2010 (UTC)


 * I would suggest to refer to the fact that different physical phenomena (actually not only physical ones, as they can also be from other fields like economy...) can have the same mathematical model. PDE's and their solutions are mathematical models for any kind of phenomena. 193.247.250.59 (talk) 21:45, 14 June 2010 (UTC)

Assessment comment
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)

Needs rewriting
The sentence:

"This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. "

neglects to state that the solution approaches infinity as what approaches what. Mathematical functions can approach infinity as certain values of their arguments approach certain limiting values. What the limiting value(s) are is unfortunately left unstated here.

It is very important in an encyclopedia article to avoid assuming that the reader is telepathic.50.205.142.50 (talk) 13:37, 27 March 2020 (UTC)

Definition of the quasi-linear equations
I saw that in the article Method of characteristics is mentioned a quasi-linear type of equation. But the definition can't be found on Wikipedia and there's a link to wolfram Mathematica language specification. I think this article could be suited for the definition of such equations. What do you think about it? Where should it be written?

Diagramlemma (talk) 15:10, 19 June 2020 (UTC)


 * If someone decides to add the definition, I have a book that can be used as a reference. MaxwellMolecule (talk) 19:13, 19 June 2020 (UTC)


 * I have made this addition, see semilinear and quasilinear in the page Nick Mulgan (talk) 04:25, 27 November 2020 (UTC)