A boundary value problem discussion

[radou]

A boundary value problem "discussion"

So, let's say we are given a function f : [0, 1] --> R and constants a, b, and we want to find u : [0, 1] --> R such that u''(x) + f = 0 on <0, 1> with u(1) = a and u'(0) = -b.

One can easily obtain the exact solution to this problem merely by using direct integration. But, my book says: "We are interested in developing schemes for obtaining approximate solutions to this problem that will be applicable to much more complex situations in which exact solutions are not possible."

Well, I'm interested in what kind of "complex situations" the author was referring to here. Is it simply the case when the given function f is not integrable?

Although this post only demonstrates my lack of knowledge in analysis, I'm still curious about it, since I want to be fully motivated to start doing some finite element "research".

Thanks in advance.

 

[dhris]

Not all differential equations can be integrated this way. In fact, most equations that you will encounter in physical situations cannot be. For those, some kind of numerical scheme is necessary. Your equation is indeed very simple, and so is ideal for discussing approximate methods that will work on more complicated equations.

 

[radou]

Not all differential equations can be integrated this way. In fact, most equations that you will encounter in physical situations cannot be. For those, some kind of numerical scheme is necessary. Your equation is indeed very simple, and so is ideal for discussing approximate methods that will work on more complicated equations.

I understand, but still, I'd like to see one of these complicated equations.

Perhaps I should just read, and sooner or later I'll get the point.

 

[symbolipoint]

I understand, but still, I'd like to see one of these complicated equations.

Perhaps I should just read, and sooner or later I'll get the point.

Wow! I just checked into this topic for variety; I know very little about differential equations, since it was so very long ago that I studied a few in an introductory course only. Your comment question reminds me of why younger students learning to use graphing calculators are usually given fairly simple problems to use for learning about the graphing calculator. You use the easy problems so that you can easily check the correctness of the result. Sure, you know how to solve the problem using the standard or easy way; but the objective now is to learn to use a DIFFERENT method or technique. Once you try the new method or technique, you can check your result by using your more familiar way.

 

[DiffEQprof]

For some helpful reference material on boundary value problems, try this book:

http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&rd=1&item=120156979964&ssPageName=STRK:MESE:IT&ih=002