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Advanced nonlinear integro-differential equation

• To: mathgroup at smc.vnet.net
• Subject: [mg71044] [mg70869] Advanced nonlinear integro-differential equation
• From: Robert Berger <rberger06 at sinh.us>
• Date: Mon, 6 Nov 2006 02:52:52 -0500 (EST)

Dear Mathematica experts! :-)

At the moment I'm dealing with the following nonlinear
integro-differential equation arising from a quantum
mechanic problem

y''[x] + 2 y'[x]/x + 2 y[x] = A (1 + B/x) y[x] f[x]

where

f'[x] = x^2 y[x] .

If the right sight of the equation is small, e.g., A = 0,
then the solution (linearized theory) is

y[x] = (C1 Sin[Sqrt[2] x] + C2 Cos[Sqrt[2] x])/x .

However, the problem is that in my case is A=1E-36, B=7.3E-3,
and y[0] = 1E26 (!) and therefore the nonlinear term is not
negligible. :-(

In this conjunction I have the following two questions:

1. Three boundary conditions are necessary.
It is easy to introduce y[0] = 1E26 and y'[0] = 0 as
boundary conditions in NDSolve but how can I use the
additional condition f[Infinity] = A?

2. It seems that the large y[0]-value cause some serious
numerical problems. Has anyone some tips, links, etc.
how to the rid of these problems?

Thanks!

Kindly regards,
Robert.