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MathGroup Archive 2006

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Re: DSolve fails with Telegraph equation

  • To: mathgroup at smc.vnet.net
  • Subject: [mg69838] Re: DSolve fails with Telegraph equation
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Mon, 25 Sep 2006 03:52:54 -0400 (EDT)
  • Organization: The University of Western Australia
  • References: <ef2tha$mmi$1@smc.vnet.net>

In article <ef2tha$mmi$1 at smc.vnet.net>,
 Oliver Friedrich <xoliver.friedrich at tzm.dex> wrote:

> I try to solve the telegraph equation
> 
> dxdx(u[x,t])==A*(u[x,t])+B*dt(u[x,t])+C*dtdt(u[x,t])
> 
> but DSolve returns immidiately without solution. I thought that this 
> equation is one of the more easy to crack for Mathematica. Am I wrong?

First, it is a bad idea to use single-letter capitals as variables. 
Second, _which_ solution to the telegraph equation were you expecting? 

Writing the equation in Mathematica notation as

  eq = D[u[x, t],{x, 2}] == 
    b D[u[x, t], t] + c D[u[x, t], {t, 2}] + a u[x, t]

then one can find traveling-wave solutions via the substitution

  eq /. u -> Function[{x, t}, f[k x - w t]] /. k x - w t -> z

and

  DSolve[%, f, z]

Alternatively, the substitution

  eq /. u -> Function[{x, t}, Exp[-b t/(2 c)] v[x, t]] 

(http://eqworld.ipmnet.ru/en/solutions/lpde/lpde207.pdf) leads to the 
Klein­Gordon equation. Separable solutions to this equation are given at

  http://eqworld.ipmnet.ru/en/solutions/lpde/lpde203.pdf

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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