       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:

>
> (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

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

```

• Prev by Date: Re: neat sums and pattered randomness
• Next by Date: RE: Re: Re: 64 bit cpu and Mathematica on windows.
• Previous by thread: Help in solving PDF equations
• Next by thread: NMinimize options