Talk:Initial value problem

Organization

Hello. It's not clear to me how this article & the other articles on diff eqs should be organized. One approach would be to put a short defn here, and then link to the main diff eq article. However, it might be neater to have the main page give an overview of several topics, one of which is the IVP, and then hand it off to this page, and put a detailed treatment of the IVP here. Maybe someone would like to comment on this question. Happy editing, Wile E. Heresiarch 04:06, 2 Mar 2004 (UTC)

Generally, I would say that differential equations is such a large topic that we should be selective in what we put in the Differential Equation article. Specifically, I am wondering what you want to say about IVPs. The only thing that comes to my mind is that they have a unique solution, which is such an important property that it should be in the Differential Equation page, in my opinion. By the way, nice work on the ODE-related pages. -- Jitse Niesen 15:34, 4 Mar 2004 (UTC)

I have changed some things that I think are important on the definition. An IVP for a DE is in fact, more generally, a relation between a function in $\mathbb {R} \times \mathbb {R} ^{n}\,$ and its derivatives with respect to time $t\,$ , actually $f\,$ is defined on an open set, here we see just the one-dimensional case, I changed it to n-dimensional case, I don't think engineers have any problem to change n by 1 to understand the same problem in the one-dimensional case :). Another thing is that the Picard's existence and uniqueness of solutions of the Cauchy's Problem or IVP states that if $f\,$ is continious and Lipschitz with respect to $y\,$ then we have existence and uniqueness in a neighbourhood of $t_{0}\,$ which actually can be found explicitly. It is not necessary to claim any kind of smoothness on $f\,$ ! Of course smoothness on $y\,$ implies Lipschitzian condition with respect to $y\,$ . —Preceding unsigned comment added by Fouri87 (talk • contribs) 11:01, 29 June 2010 (UTC)[reply]

Difficulties and Confusion

Great job on the article writers on making a text book example of a text book definition. How many articles are there on the internet that make a simple concept so complex and scientific so that the average person goes "wow, I hate math, maybe I should pick a degree in Business or Writing". Articles like these contribute to the lowering of the number of engineers, mathematicians, scientists around the world. When will you guys understand you have to think with a newbie head to teach a newbie. At least people in the programming and web development field have figured out ways to make things so much more simpler to explain.

Your article should contain simple and effective language and a couple mathematical examples with solutions that are not similar to a text book that seems to skip steps and make things look complicated.198.82.127.163 03:55, 10 September 2007 (UTC)[reply]

Worked example

I came to this page after seeing the following problem [1] and agree with the anonymous editor above that this article needs simpler and more effective language. I have added a simple worked example of the 'Fail Blog' problem, following the method at [2]. —Preceding unsigned comment added by BruceMcAdam (talk • contribs) 15:01, 28 October 2009 (UTC)[reply]

That means what did you do on bayram holiday? —Preceding unsigned comment added by 193.140.249.2 (talk) 15:38, 20 May 2010 (UTC)[reply]

Cauchy problem

As far as I know, a Cauchy problem is an initial value problem on unbounded domains with vanish-at-infinity conditions. Is anyone familiar with authors who use the term for any initial value problem? — Preceding unsigned comment added by Dingenis (talk • contribs) 12:53, 12 April 2012 (UTC)[reply]

"Exponential smoothing"

I am removing this text because it has nothing to do with initial value problems in ODE theory, and it is probably only here due to somebody misunderstanding the title. I will also copy the text to the Exponential smoothing talk page. --138.38.106.191 (talk) 14:22, 10 May 2013 (UTC)[reply]

Exponential smoothing is a general method for removing noise from a data series, or producing a short term forecast of time series data.

Single exponential smoothing is equivalent to computing an exponential moving average. The smoothing parameter is determined automatically, by minimizing the squared difference between the actual and the forecast values. Double exponential smoothing introduces a linear trend, and so has two parameters. For estimating initial value there are several methods. like we use these two formulas;

y'_{0}=\left({\frac {\alpha }{1-\alpha }}\right)a_{t}+b_{t}

y''_{0}=\left({\frac {\alpha }{1-\alpha }}\right)a_{t}+2b_{t}

Problematic text about Existence and Uniqueness from numerical calculation

The text at the start of existence and uniqueness currently (2013 Jun 25) claims that: "For a large class of initial value problems, the existence and uniqueness of a solution can be illustrated through the use of a calculator." This seems misleading and false. Many numerical methods will return "a solution," even if the integration method is unstable. Also, numerical representations are inherently inaccurate at some level of precision, so establishing uniqueness numerically is problematic. — Preceding unsigned comment added by 50.131.141.62 (talk) 18:33, 25 June 2013 (UTC)[reply]

The introductory paragraph is confusing!

[I'm posting this because I think the language in the opening paragraph of pages should be tailored to people who do not yet know what the page is about. I feel the opening paragraph on this page should be improved by someone who understands this topic. I'm posting my thoughts while do not understand the topic in the hopes that my rambling thoughts will help someone who can edit the page in the future. This is not rigorous criticism but confused impressions.]

With a basic background in mathematics (first year university) the opening paragraph is pretty confusing. It sounds like describing Hunger as:

Hunger is a human condition.
Eating takeout is a common way of removing hunger.
Takeout is normally ordered by phone.

What is an Initial Value Problem when it's not solved by modelling a system? Outside of "that context" what is an Initial Value Problem? What's the purpose / aim / goal of having this mathematical construct? Why is it useful?

The first sentence is (paraphrasing):

An initial value problem is:

an ordinary differential equation

an initial condition

But both an equation and an initial condition are constraints while 'problem' implies a question - what's the question? "ODE + initial condition" seems like a necessary condition but not sufficient? If it's a class of problems it should say this explicitly, like "An IVP is any problem where XYZ".

I don't know how you would rewrite the paragraph to make more sense, but as it is right now it's unclear. - 2A00:23C5:E406:3700:B8C3:B42C:3CCD:32E0 (talk) 01:01, 22 July 2020 (UTC)[reply]

The problem is to "solve the equation" - This can mean different things as there are analytical solutions and numerical solutions. Discussion of what a solution is may be beyond the scope of the page. An initial value problem often specifies constraints when t=0. This is often contrasted with a "boundary value problem". — Preceding unsigned comment added by 161.11.160.176 (talk) 18:21, 23 December 2024 (UTC)[reply]

Vandalism

For some reason there seems to be an unusually large amount of vandalism. Should the page be protected temporarily? --James Az. H (talk) 15:30, 27 August 2020 (UTC)[reply]

Numerical vs analytical solutions

The current article focuses exclusively on analytical solutions despite such solutions not always existing in elementary form. Generally speaking numerical solutions of IVP's are probably at least as commonly discussed as analytical solutions. Jason Quinn (talk) 18:08, 13 October 2023 (UTC)[reply]