Re: Questionable solution from DSolve
- To: mathgroup at smc.vnet.net
- Subject: [mg63679] Re: Questionable solution from DSolve
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Mon, 9 Jan 2006 04:49:58 -0500 (EST)
- Organization: The University of Western Australia
- References: <dpqjr1$2hq$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <dpqjr1$2hq$1 at smc.vnet.net>, dkjk at bigpond.net.au wrote:
> Consider the differential equation
>
> D[f[r,z], {r, 2}] + (1/r)D[f[r, z], {r, 1}] - (kz^2 + 1^2)f[r, z] ==
> g[r,z]
>
> With the RHS set to 0, the solution is simply
>
> BesselJ[0, I*r*Sqrt[kz^2+1]]C[1][z] + BesselY[0,
> -I*r*Sqrt[kz^2+1]]C[2][z]
>
> as expected. But if I put g[r,z] = - Exp[-a * r^2 - b * z^2]
>
> I obtain
>
> BesselJ[0, I*r*Sqrt[kz^2+1]]C[1][z] + BesselY[0,
> -I*r*Sqrt[kz^2+1]]C[2][z]
>
> plus some integral with 1 in the lower terminal and r in the upper
> terminal. This is very odd, since the solution is not even
> dimensionally correct! The variable r can be considered to have units
> of metres whereas unity is dimensionless. You might claim that the
> original differential equation is not dimensionally corect to begin
> with, but multiply any of the terms by arbitrary constants and you
> still end up with unity in the lower terminal.
There is nothing wrong with the solution returned by Mathematica --
except that it is not in simplest form. You can always re-scale your
variables to make the original equations dimensionless.
However, DSolve does not reduce expressions involving the Wronskian of
the solutions to a give differential equation. In this particular case,
Mathematica does not recognize that
BesselI[1, t] BesselY[0, -I t] - I BesselI[0, t] BesselY[1, -I t]
is just
-2/(Pi t)
which greatly simplifies the integrals returned by NDSolve.
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
School of Physics, M013 Fax: +61 8 6488 1014
The University of Western Australia (CRICOS Provider No 00126G)
AUSTRALIA http://physics.uwa.edu.au/~paul