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