       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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