1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Leading order matched

  1. Sep 21, 2008 #1
    What would be the leading order match asymptotic expansion for the following differential equation?
    [tex]\epsilon y'' = f(x) - y' [/tex]where f(x) is continuous, [tex]\epsilon<<1[/tex]
    and y(0) = 0, y(1) = 1

    Thanks in advance,
  2. jcsd
  3. Sep 21, 2008 #2


    User Avatar
    Science Advisor

    First, what is an "asymptotic expansion", both in general and for this particular problem? Then what is meant by a "leading order match"?
  4. Sep 22, 2008 #3
    Firstly, thank you for your response.
    To the best of my knowledge, an asymptotic expansion is in general defined this way:

    1. First, an asymptotic sequence is formed from functions called scale, or gauge, or basis functions, denoted [tex]\phi_{1}\phi_{2},...[/tex]. These functions are well-ordered, which means that [tex]\phi_{n}=o(\phi_{m}) [/tex] as some [tex]\epsilon[/tex] (epsilon)gets really really small for all m and n that satisfy m<n

    2. Now, if [tex]\phi_{1}\phi_{2},...[/tex] is an asymptotic sequence, then f(epsilon) has an asymptotic expansion to n terms, with respect to this sequence, if and only if

    [tex]f = {^m}\sum_{k=1} a_{k}\phi_{k}(\epsilon)+o(\phi_{m})[/tex] for m=1,...,n as epsilon gets really really small towards like 0. The [tex]a_{k}[/tex] are independent of [tex]\epsilon[/tex].

    Finally, all this enables us to write f~[tex]a_{1}\phi_{1}(\epsilon)+a_{2}\phi_{2}(\epsilon)+...+a_{n}\phi_{n}(\epsilon) [/tex] as [tex]\epsilon\rightarrow 0[/tex]. Here, the ~ denotes asymptotic.

    Whew! Now, for this particular problem... I think the procedure is to find the outer solution away from the boundary layer (reduce given original equation by setting [tex]\epsilon[/tex] = 0), and then the inner solution near the boundary layer (in this case near x=0) and then push both towards each other into an overlap region in which they are supposed to match!

    I have started on the outer problem thus far:
    y' = f(x) so y(x) = [tex]\int f(x)dx[/tex] and now what do I do with this integral?

    Thanks again!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook