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Re: Advanced nonlinear integro-differential equation


Robert Berger schrieb:
> 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?

Set
eq[a_,b_,c_,d_,e_] := {
     y''[x] + 2 y'[x]/x + 2 y[x] == a(1 + 2/x) *y[x] *f[x],
     f'[x] == x^2 y[x], f[1] == 1, y[1] == 1, y'[1] == 1}


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