Asymptotic mathcing for a first order differential equation

[hanson]

The first-order differential equation
$$y' +(ex^2+1+1/x^2)y=0$$, with boudary value y(1) =1

Using, asymptotic mathcing to study the behaviour of the sltion as e tends to +0, when x is not too large, the term $$ex^2$$ is negligible so an approximate equation for y is
$$y'_L +(1+1/x^2)y_L=0$$.

When x is large, [ tex ]ex^2 [ /tex ] is not negligible but $$1/x^2$$ is. Therefore, an approximate equation valid as x tends to infinity is $$y'_R +(ex^2+1)y_R=0$$.

I think that te upper edge of the left region would be the largest value of x for which ex^2 is still small compared with 1. This would suggest that the left region consists of those x for which x<<e^-(1/2) as e tends to +0. But actually the region of validity of the left solution is e^(-1/3) (e tends to +0)...Can anyone explain this to me??

Sams as the right region...Please kindly help

 

[Clausius2]

That's a nice quote of the example lying in Benders&Orszag book. It's a nice book, but sometimes is pretty tough to follow. And I remember this same question arised in class when we saw this example.

Actually the interval of matching for x seems diferent at leading order than that foresought by looking at the equation. The matching region is not given by the approximation you make on the terms of the equation but is given from the matching itself.

The left solution is $$y_L=e^{-x+1/x}+O(\epsilon)$$ which for large $$x$$ gives $$y_L\sim e^{-x}[1+1/x+O(1/x^2,\epsilon)]$$.

On the other hand, the right solution is $$y_R=a e^{-\epsilon x^3/3-x}$$, which for small $$x$$ gives $$y_R\sim e^{-x}[1-\epsilon x^3/3+O(\epsilon^2 x^6)]$$.

That is, for $$x>>1$$ AND $$x<<\epsilon ^{-1/3}$$ (look at the second term on the right solution expansion), then both approximations have the same functional dependence, i.e. $$y\sim e^{-x}$$ if $$a=1$$. The uniform approximation to leading order is then $$y=y_R+y_L-e^{-x}$$.

Successive approximations will probably narrow the matching region.

I thought you were doing undergrad, this is a little advanced for it, isn't it?. Hope you are getting interest on this matter, a difficult one though. For me it sounds challenging, my thesis is all about this applied to Fluid Mechanics theory.

 

[hanson]

Clausius2! Thank you so much! I have waited for the answer to this question for long!

Yes..that book is pretty tough to follow...particularly for an undergraduate student like me, who is a bit crap in mathematics...

Is there any other books that you think may act as reference or some sort of help so that I can follow Bender a bit easier?

 

[Clausius2]

Clausius2! Thank you so much! I have waited for the answer to this question for long!

Yes..that book is pretty tough to follow...particularly for an undergraduate student like me, who is a bit crap in mathematics...

Is there any other books that you think may act as reference or some sort of help so that I can follow Bender a bit easier?

The answer is no as far as i know. I was taught based on that book, but it was a graduate level course. Asymptotic methods is something used typically in grad level, and I use it for working out approximate analytical solutions to moderately complex fluid flows. What one does is to identify a small parameter (Reynolds#, a ratio of dimensions, a ratio of times), called the perturbation parameter, and solve for a solution in series of that small parameter (an asymptotic series). I do know that for instance WKB method is also used widely in Quantum Mechanics.