Re: Advanced nonlinear integro-differential equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg71060] Re: Advanced nonlinear integro-differential equation*From*: Roland Franzius <roland.franzius at uos.de>*Date*: Wed, 8 Nov 2006 06:01:34 -0500 (EST)*Organization*: Universitaet Hannover*References*: <ei4lej$dfq$1@smc.vnet.net>

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}