Talk:Boundary value problem

Examples
I think the worked examples are overkill. This page should be brief, and the worked examples moved to more specific topics (Strurm-Liouville problem or elliptic eigenvalue problems as the case may be). Brian Tvedt 14:11, 14 August 2005 (UTC)


 * Hi Brian, welcome here. Why do you say that this page should be brief? It seems quite a good idea to list some examples. However, the current examples are probably not the best illustration, because they are eigenvalue problems which complicates matters. I think it's better to drop the parameter &lambda;. By the way, have you already come across WikiProject Mathematics? I hope that you decide to stay around and improve some of our articles. Cheers, Jitse Niesen (talk) 15:14, 14 August 2005 (UTC)
 * I very much agree with Jitse. Brian, of the bad things about the math articles on Wikipedia is that sometimes they tend to be written by mathematicians for mathematicians who already know the thing. As such, a lot of math articles are completely incomprehensible to nonmathematicians.
 * According to the math style manual, at least the introduction of an article should be at the level of high school students. Thus, examples are definitely helpful. A dry definition willl not help anybody learn. Oleg Alexandrov 15:37, 14 August 2005 (UTC)
 * Oleg, I'm certainly not saying the article should be overly technical--quite the contrary! I think it would be improved by more emphasis on high-level description, rather than grinding through separation of variables.  I added an (overly simplistic) paragraph on the notion of well-posedness as a nudge in this direction.
 * To answer Jitse's question, part of my thinking was that we already have a subtopic in the PDE article on specific methods of solution, and that would be a better place to collect worked examples. I'm certainly not opposed to examples in principle.  (BTW, the kind welcome is appreciated.) (unsigned post by Brian Tvedt)

I see your point. Your high-level description coupled with my simple example (without separation of variables) might as well make people understand what is going on. I still like the PDE example though, even if it is technical. It is the last example, so it can be complicated, and it gives a good idea of where boundary value problems are used. Oleg Alexandrov 00:35, 16 August 2005 (UTC)

as a non mathematician...
I can't get a mastery of the subject, but I wonder, is it possible that it bears on the problem of the 'joints' as mentioned by Socrates in the Symposium? That is, our senses allow us to perceive certain kinds of boundries intuitively but how do we decide where the joints lie otherwise. I know this is a very long shot ( in the dark ) but perhaps someone will help me find a matchup somewhere in maths. wblakesx


 * I'm not familiar with the passage, and don't have Plato handy, but it is conceivable that Socrates was fumbling towards a concept related in a crude way to the modern mathematical notion of 'boundary value problem'. Such a connection would be fascinating, but if you find it, keep in mind this counts as original reseach, so Wikipedia is not the place to publish it. Brian Tvedt 12:50, 22 December 2005 (UTC)

Other types of boundary value problems
There are many boundary value problems studied by mathematicians (mostly complex analysts and potential theorists) which do not fall under the heading of solutions of differential equations. For example, there is a lot of research into the boundary behaviour of (discontinuous) subharmonic functions, meromorphic functions, fine-continuous functions, etc. etc. I feel tempted (when I get some time) to write something about these problems somewhere, but the question is: do they belong here, or in a separate article, and if so, then what should it be titled? Madmath789 09:10, 1 August 2006 (UTC)


 * I would suggest to start a section here and if it is too big create a separate article. Rex the first  talk 23:00, 1 August 2006 (UTC)


 * Subharmonic functions solve an equation $$\Delta u = f$$ with $$f<0$$, which satisfy certain maximum principles (see elliptic boundary value problem), this is still a PDE problem. Meromorphic functions satisfy the Cauchy-Riemann differential equation (a first order vector equation) so again, this is a PDE problem. Furthermore, the real part of a meromorphic function is harmonic, and the imaginary part is its "harmonic conjugate" (so again, see elliptic boundary value problem.) I don't know about fine continuous functions. Loisel 00:23, 18 April 2007 (UTC)

Diagrams
I have a couple of 2D Heat Equation and Laplace Equation PDE worked examples and graphs. One of which I put on the Heat equation page a while ago. Here and here. Feel free to use them on this page, if you think they might be appropriate. --Wtt 22:21, 17 April 2007 (UTC)

Boundary conditions outside of mathematics
I'm struggling (briefly) to consolidate the difference between boundary conditions in relation to boundary value problems and as the term is used in engineering. Where I've come to is that, while about the same thing, and while the physics dictating the equations used may involve BVPs, engineers tend to use the term much more generically - any set of inputs to some set of calculations are deemed to be the boundary conditions.

This may just be a local (mis)use of the term, and someone who has a better grasp on the concept from a mathematical background may be able to offer up a different POV. Just wondering if there's grounds for a short section on the other uses of the term, a Boundary Conditions (Engineering) page (though I don't see it growing beyond a stub,) or some other solution. Marimvibe (talk) 22:53, 20 June 2008 (UTC)

Special cases
It could be helpful to discuss various special cases in subsections. As it stands, the article is sometimes unclear about how general the concept of a boundary value problem is. The graph at the top shows a closed boundary in two-dimensional space. But parts of the text seem to take boundary value problem to mean specifically two-point boundary value problem.

In particular, there is a discussion of how initial value problems differ from boundary value problems, which states that the former only take conditions at t=0 as given, whereas the latter take as given conditions at t=0 and t=1. I think it should be made clear that this is just one example of a particular class of boundary value problems. Discussing two-point problems in detail, and then stating that there are many more general boundary value problems (like the one illustrated at the top of the page) would be helpful.

A distinction on this page between initial value problems and two-point boundary problems would be especially helpful for me as an editor of economics articles, because it's a crucial analytical difference in macroeconomics. Macroeconomic models based on adaptive expectations are generally initial value problems, because decision-makers' expectations in those models are determined by past experience, whereas macroeconomic models based on rational expectations are generally two-point boundary problems, because decision-makers' expectations in those models are determined by reasoning about the future. Also, from an economist's point of view, it would be helpful if this article mentioned that the distinction between initial value problems and two-point boundary value problems arises for difference equations as well as ordinary differential equations. In other words, it applies to dynamic systems in general, not to continuous-time dynamics only.

Any objections if I create a subsection specifically on two-point boundary problems? --Rinconsoleao (talk) 18:30, 2 March 2009 (UTC)

Another point: the introduction defines a boundary value problem as a differential equation together with a boundary condition. I would agree with that definition. But the subsection on initial value problems appears to say that initial value problems and boundary value problems are two different things. To my understanding that's incorrect. The overall definition of boundary value problems would imply that initial value problems are a special case of boundary value problems. --Rinconsoleao (talk) 18:37, 2 March 2009 (UTC)

…4:11 P.M. E.S.T. Could this be an Initial ValueDavid George DeLancey (talk) 21:12, 6 November 2009 (UTC) I researched word term ENTITY there was no related date as to when the word and or term was invented. Thank You Keep In Touch.David George DeLancey (talk) 21:12, 6 November 2009 (UTC)

Normal derivative vs. temperature
There are currently these examples provided in this article, presumably hinting on the heat equation:
 * If the boundary gives a value to the normal derivative of the problem then it is a Neumann boundary condition. For example, if there is a heater at one end of an iron rod, then energy would be added at a constant rate but the actual temperature would not be known.
 * If the boundary gives a value to the problem then it is a Dirichlet boundary condition. For example, if one end of an iron rod is held at absolute zero, then the value of the problem would be known at that point in space.

Thanks, --Abdull (talk) 16:56, 15 February 2011 (UTC)
 * In the Neumann example, what does the heat equation have to do with the normal derivative? What does a heater at one end of an iron rod have to do with a normal derivate?
 * Conversely, what makes holding the rod at absolute zero different from placing a heater at this place, such that we don't work with the normal derivative?

Boundary condition in mathematical induction
In Thomas H. Cormen's Introduction to Algorithms (3rd edition) p84, there's the following passage:
 * We can use the substitution method to establish either upper or lower bounds on a recurrence. As an example, let us determine an upper bound on the recurrence T(n)=2T(n/2)+n:
 * Mathematical induction now requires us to show that our solution holds for the boundary conditions. Typically, we do so by showing that the boundary conditions are suitable as base cases for the inductive proof.
 * Mathematical induction now requires us to show that our solution holds for the boundary conditions. Typically, we do so by showing that the boundary conditions are suitable as base cases for the inductive proof.

Does the definition of "boundary condition" as found on this article help me understand the use of "boundary condition" in this sentence? The article refers to it as a term in differential equations, which is not what Cormen's books deals with. --ToastieIL (talk) 16:11, 27 November 2011 (UTC)

Finite Difference Method in Numerical Section
Why does the 'numerical solutions' section not mention finite difference methods? You should be able to choose between shooting method variations or finite difference methods to solve an arbitrary boundary value problem. Wiki47222 (talk) 00:42, 28 March 2019 (UTC)