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Rewrite needed

Can somebody who understands this article rewrite the following please? "Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability." As is, it is incomprehensible. Perhaps break it into two or three sentences? — Preceding unsigned comment added by 2600:1702:170:3B90:252F:91F:542:7612 (talk) 17:59, 26 October 2022 (UTC)[reply]

I agree that area is incomprehensible. I find the entire article to be incomprehensible. Start over. Boil it down to exactly what is a PDE. Precisely. Explain each part precisely. Only then go into why PDE's are useful, and solutions and etc..

J Mark Morris (talk) 01:44, 20 May 2024 (UTC)[reply]

Even the first sentence is very confusing: "In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function." I cannot tell what the word "between" is supposed to mean there, and an equation doesn't "compute" anything. 74.96.10.88 (talk) 15:01, 10 December 2024 (UTC)[reply]

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) Tianran, are you still alive? — Preceding unsigned comment added by 37.232.69.137 (talk) 04:17, 19 April 2022 (UTC)[reply]

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.

Wiki Education assignment: 4A Wikipedia assignment

This article was the subject of a Wiki Education Foundation-supported course assignment, between 13 February 2023 and 12 June 2023. Further details are available on the course page. Student editor(s): Jalenlthomas (article contribs). Peer reviewers: Ogomzn, Jmorales169.

— Assignment last updated by Lzepeda12 (talk) 21:08, 26 April 2023 (UTC)[reply]

Comparison with simple differential equations

It would be nice to have an explanation of the difference between simple differential equations in one variable and partial differential equations in multiple variables, with worked examples. The current article tells us how they are used but not the very basic step of what happens to the other variables, an explanation of which would let people take the step between the two more easily. Robin Murison (talk) 10:58, 20 April 2023 (UTC)[reply]

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-== Bad Mathematics ==
+| maxarchivesize=75K
-This article is flawed from the onset. I tried to give it some rigour, but just gave up. For example: If <math>u_x = 0</math> then <math>u(x,y)</math> is independant of <math>x</math>, and so <math>u(x,y) = f(y)</math> "where <math>f</math> is an arbitrary ''(differentiable)'' function of <math>y</math>..." Well, try <math>u(x,y) = |y|</math>. Clearly <math>u_x = 0</math> for all <math>x</math>, but <math>|y|</math> is discontinuous at <math>y = 0</math>, and hence so too is <math>\partial |y| / \partial y</math>.
+| archiveheader={{Automatic archive navigator}}
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+==Rewrite needed==
+Can somebody who understands this article rewrite the following please?  "Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability."  As is, it is incomprehensible.  Perhaps break it into two or three sentences?  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2600:1702:170:3B90:252F:91F:542:7612|2600:1702:170:3B90:252F:91F:542:7612]] ([[User talk:2600:1702:170:3B90:252F:91F:542:7612#top|talk]]) 17:59, 26 October 2022 (UTC)</small> <!--Autosigned by SineBot-->
+:I agree that area is incomprehensible. I find the entire article to be incomprehensible. Start over. Boil it down to exactly what is a PDE. Precisely. Explain each part precisely.  Only then go into why PDE's are useful, and solutions and etc..
-:But <math>u(x,y)=|y|</math> IS continuous everywhere. [[User:Saran T.|Saran T.]] ([[User talk:Saran T.|talk]]) 12:24, 21 April 2008 (UTC)
+:[[User:J Mark Morris|J Mark Morris]] ([[User talk:J Mark Morris|talk]]) 01:44, 20 May 2024 (UTC)
+::Even the first sentence is very confusing: "In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function." I cannot tell what the word "between" is supposed to mean there, and an equation doesn't "compute" anything. [[Special:Contributions/74.96.10.88|74.96.10.88]] ([[User talk:74.96.10.88|talk]]) 15:01, 10 December 2024 (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 <math>f(y)</math> is not differentiable.
-::[[User:Thenub314|Thenub314]] ([[User talk: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. -- [[User:Jitse Niesen|Jitse Niesen]] ([[User talk:Jitse Niesen|talk]]) 10:14, 22 April 2008 (UTC)
 == Notation and examples ==
@@ Line 22: / Line 28: @@
 '''thermal diffusivity'''.<br>
 [[User:Chentianran|Tianran Chen]] 03:56, 2004 Mar 1 (UTC)
+Tianran, are you still alive?  <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/37.232.69.137|37.232.69.137]] ([[User talk:37.232.69.137#top|talk]]) 04:17, 19 April 2022 (UTC)</small> <!--Autosigned by SineBot-->
-== 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.<br>
-[[User:Chentianran|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.--[[User:Karl-H|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.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/75.173.40.108|75.173.40.108]] ([[User talk:75.173.40.108|talk]]) 22:05, 21 May 2011 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
-== Classification ==
-I don't understand the section on classification into hyperbolic, etc.  Where is this <math>\!\, v_t-v_x</math> 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 <math>B^2-2AC</math> with ''B'' the coeffecient of <math>u_{xy}</math> and so on. --[[User:Ub3rm4th|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. --[[User:Eienmaru|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. [[Special:Contributions/193.247.250.59|193.247.250.59]] ([[User talk:193.247.250.59|talk]]) 21:37, 14 June 2010 (UTC)
-== 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.
-:--[[User:Gseryakov|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. &ndash;[[User:Matthew Low|Matt]] 07:42, 3 November 2005 (UTC)
-== First order system example removed ==
-:''The matrix associated with the system''
-::<math>u_t+2v_x=0</math>
-::<math>v_t-u_x=0</math>
-:''has coefficients,''
-::<math>\begin{bmatrix}
-& 0 \\
-& -1\end{bmatrix}</math>
-:''The eigenvectors are (0,1) and (1,0) with eigenvalues 2 and -1. Thus, the system is hyperbolic.''
-Some problems with this:
-# The conclusion is wrong.  Eliminating ''v'' from the system we find that ''u'' satisfies <math>u_{xx} + 2 u_{yy} = 0</math>, meaning the system is elliptic, not hyperbolic.
-# It is not at all clear how the matrix is related to the coefficients of the system.
-# 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:
-:<math> A_{11} u_x + A_{12} u_y + B_{11} v_x + B_{12} v_y = 0 </math>
-:<math> A_{21} u_x + A_{22} u_y + B_{21} v_x + B_{22} v_y = 0 </math>
-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 <math>det(A-\lambda B)=0</math>. [[User:Brian Tvedt|Brian Tvedt]] 02:54, 23 November 2005 (UTC)
 == Conceptual Definition ==
@@ Line 108: / Line 41: @@
 What i would like to know is if there exists such a nice conceptual definition of a PDE.
+==Wiki Education assignment: 4A Wikipedia assignment==
-== Bad Mathematics? ==
+{{dashboard.wikiedu.org assignment | course = Wikipedia:Wiki_Ed/Riverside_City_College/4A_Wikipedia_assignment_(spring_2023) | assignments = [[User:Jalenlthomas|Jalenlthomas]] | reviewers = [[User:Ogomzn|Ogomzn]], [[User:Jmorales169|Jmorales169]] | start_date = 2023-02-13 | end_date = 2023-06-12 }}
-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.<br />
-Mentioned in another part of the Wikipedia is the concept of weak solutions.  Weak solutions play an important role in the theory of PDEs. --[[User:Jss214322|Jss214322]] ([[User talk:Jss214322|talk]]) 00:54, 2 February 2008 (UTC)
-== Intro paragraph question ==
-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''?  [[User:Rhetth|Rhetth]] ([[User talk: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. [[Special:Contributions/193.247.250.59|193.247.250.59]] ([[User talk:193.247.250.59|talk]]) 21:45, 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  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/203.206.255.53|203.206.255.53]] ([[User talk:203.206.255.53|talk]]) 15:05, 20 July 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->
-: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. -<small>[[User:PrBeacon|PrBeacon]] [[User_talk:PrBeacon|(talk)]]</small> 07:51, 6 October 2010 (UTC)
-== Spectral Methods ==
-Is another important method to solve PDEs (numerically). --[[User:Jorgecarleitao|Jorgecarleitao]] ([[User talk:Jorgecarleitao|talk]]) 08:46, 2 November 2011 (UTC)
+<span class="wikied-assignment" style="font-size:85%;">— Assignment last updated by [[User:Lzepeda12|Lzepeda12]] ([[User talk:Lzepeda12|talk]]) 21:08, 26 April 2023 (UTC)</span>
+== Comparison with simple differential equations ==
-== Inline citations missing? ==
+It would be nice to have an explanation of the difference between simple differential equations in one variable and partial differential equations in multiple variables, with worked examples.
-There are 19 references for the article. There does not appear to be any inline citations? --Михал Орела 14:14, 25 January 2012 (UTC)  <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:MihalOrela|MihalOrela]] ([[User talk:MihalOrela|talk]] • [[Special:Contributions/MihalOrela|contribs]]) </span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->
+The current article tells us how they are used but not the very basic step of what happens to the other variables, an explanation of which would let people take the step between the two more easily. [[User:Robin Murison|Robin Murison]] ([[User talk:Robin Murison|talk]]) 10:58, 20 April 2023 (UTC)