1. Not finding help here? Sign up for a free 30min 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!

Asymptotic Expansion of ODE

  1. Jul 9, 2013 #1
    I'm trying to approximate [itex]f'(r)[/itex] for the following equation using matched asymptotic expansions

    [itex]-\frac{1}{2}\epsilon ff''=\left[\left(\epsilon+2r\right)f''\right]'[/itex]

    where [itex]\epsilon \ll 1[/itex] and with the boundary conditions [itex] f(0)=f'(0)=0, \quad f'(\infty)=1[/itex]

    The inner expansion which satisfies [itex] f'(0)=0[/itex] is simple enough by choosing an appropriate inner variable.

    My problem is trying to form an outer expansion of the form

    [itex] f'=1+\sigma(\epsilon) f_1+ \dots [/itex]

    where [itex]\sigma[/itex] is found through matching. In my working I find [itex]f_i≈A_i\ln r[/itex] which obviously doesn't satisfy [itex]f'(\infty)=0[/itex] unless the constants equal zero.

    I've tried introducing a stretched variable of the form [itex] \gamma =\epsilon r[/itex] but with no success.

    Any suggestions?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?