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MathGroup Archive 2004

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Re: Help in solving PDF equations

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52245] Re: Help in solving PDF equations
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 17 Nov 2004 02:20:13 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <cnbn3o$9va$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <cnbn3o$9va$1 at smc.vnet.net>, Wei Wang <weiwang at baosteel.com> 
wrote:

> Could anybody please help me in solving the following PDF equation?
> 
> (p + r/rou)*(1- (Derivative[1, 0][f][x, t])^2 - k * Derivative[0, 1][f][x, 
> t])/Sqrt[1 + (Derivative[1, 0][f][x, t])^2] ==0

Clearly, the left-hand side of this expression vanishes if its numerator 
does:

 de = (1- Derivative[1, 0][f][x, t]^2 - k Derivative[0, 1][f][x, t]) == 0

and Mathematica can solve this partial differential equation in closed 
form:

 sol = DSolve[de, f, {x, t}]

> with initial or boundary conditions as
> 
> f[x, 0] == -1

Applying this initial condition

 f[x, 0] == -1 /. sol

one can find the two undetermined coefficients:

 coefs = SolveAlways[#,x]& /@ % // Union

The solution then reads

 f[x, t] /. sol /. coefs // Flatten // Union

 {t/k - 1}

which is independent of x.

Note that the denominator of the original equation reduces to unity as 
can be seen by evaluating

 1 + Derivative[1, 0][f][x, t]^2 /. sol /. coefs

> Derivative[0, 1][f][x, 0] == p/k

This condition can only be satisfied if p == 1 as can be seen from

 Derivative[0, 1][f][x, 0] == p/k /. sol /. coefs

> Derivative[1, 0][f][R0, t] == 0

This condition is satisfied since f is independent of x,

 Derivative[1, 0][f][R0, t] == 0 /. sol /. coefs

> f[x, t] == If[0<= x <=1, Sqrt[1 - x^2]]

This does not make sense to me. The right-hand side does not depend on 
t. If you mean this to hold for a fixed t then it is incompatible with 
the general solution determined above.
 
> where, p, k, R0 and rou are constants.

The supplied conditions determine p but k, R0, and rou are not 
determined.

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)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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