Matching Inner and Outer Expansions for Approximating ODE Solutions

[Chewie666]

I'm trying to approximate $f'(r)$ for the following equation using matched asymptotic expansions

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

where $\epsilon \ll 1$ and with the boundary conditions $f(0)=f'(0)=0, \quad f'(\infty)=1$

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

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

$f'=1+\sigma(\epsilon) f_1+ \dots$

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

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

Any suggestions?

 

[lippyka]

One approach you could take is to consider an inner region and an outer region. The inner region would be for r<\epsilon^{-1} and the outer region for r>\epsilon^{-1}. In the inner region, you can assume that \epsilon is small enough that it does not affect the solution much, and so you can make the approximation f' \approx 1 - A_1r^2. For the outer region, you can use an asymptotic expansion of the form f' = 1+\sigma(\epsilon)f_1 + \dots. You can then match the two solutions at r=\epsilon^{-1} to get a relation between \sigma and A_1.