Re: Imposing boundary condition at infinity

• To: mathgroup at smc.vnet.net
• Subject: [mg63266] Re: Imposing boundary condition at infinity
• From: bghiggins at ucdavis.edu
• Date: Tue, 20 Dec 2005 23:35:54 -0500 (EST)
• References: <do68nl\$b5s\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

The basic idea is to apply the BC at some finite but large r.  For the
purpose of this discussion let us write your ODE  as

Ï?''[r] + 2  Ï?[r]/r == Î± Ï?[r] + Î² Exp[-Î³ r^2]

where Î±, Î² Î³ are constants . For large r we can approximate  the
above ODE as

Ï?''[r] == Î± Ï?[r]

The general solution of the above ODE  that satisfies the BC at
infinity is

Ï?[r] = C[1] Exp[-Î±^1/2 r]

where I have assumed that Î±^1/2 is real and postive.  Differentiating
this equation gives

Ï?'[r] = -C[1]Î±^1/2 Exp[-Î±^1/2 r].

Thus for large r the following must hold

Ï?'[r] = -Î±^1/2 Ï?[r]

Hence at some finite but large r say r=r1 the original ODE must satisfy

Ï?'[r1] = -Î±^1/2 Ï?[r1]

Now you are in a position to solve your original BVP but posed on a
finite domain. Select a valuer for r1 and solve your BVP using the
above as a BC at r=r1.  Then check that the solution does not change
when r1 is increased.

Cheers,

Brian

dkjk at bigpond.net.au wrote:
> Hi all,
>
> I need to solve a differential equation whose solution has the
> contraint that it tends to zero in the infinite limit. Mathematica will
> not, however, allow \[Phi][Infinity]==0 as one of the contraints. Does
> mathematica have another notation for dealing with this constraint?
>
> Thanks.
>
> James
>
> \!\(DSolve[{\(1\/r\^2\) D[r\^2*D[Ï?[r], r],
>             r] == \(1\/Ïµ\) \((Ï?[r] \((\(2*ec^2*n0\)\/\(k*T\))\) +
> \(27*
>             q\)\/\(2*Pi*
>             a\^3*\@\(2*Pi\)\)*
>                 Exp[\(-\(\(9  r^2\)\/\(2  a^2\)\)\)])\), Ï?[0] ==
> \(\(3\ \@2\ \
> q\ Î»\)\/Ï?\^\(3/2\) - \(\[ExponentialE]\^\(a\^2\/\(18\ Î»\^2\)\)\ q\ \
> \@\(a\^2\/Î»\^2\)\ Î»\)\/Ï? + \(a\ \[ExponentialE]\^\(a\^2\/\(18\
> Î»\^2\)\)\
>                     q\ Erf[a\/\(3\ \@2\ Î»\)]\)\/
>         Ï?\)\/\(4\ a\ Ïµ\ Î»\), Ï?[Infinity] == 0}, Ï?, r]\)

```

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