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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


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