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free boundary condition

• To: mathgroup at smc.vnet.net
• Subject: [mg73306] free boundary condition
• From: "Ferdinand" <Ferdinand.Cap at aon.at>
• Date: Sun, 11 Feb 2007 01:12:52 -0500 (EST)

Hi,
could someone help to solve with MATHEMATICA the following boundary
value problem of 2 part diff equ:
1. the equations :   Laplacian[U[x,y,z,t]]=0   and D[U[x,y,z,t],t]
+0.5*Grad[U[x,y,z,t]]^2+g*S[x,y,t] =0,  g is a const.
The unknown functions are : U[x,y,z,t}  and S[x,y,t]
2.initial condition:  S[0,0,0]=0.5
3.bounday conditions:
3.1  fixed : D[U[x,y,z,t],x]*D[h[x,y],x]+D[U[x,y,z,t],y]*D[h[x,y],y]
+D[U[x,y,z,t],z]=0 at z=-h[x,y],
where h[x,y] is a numerically given function.
3.2 free Boundary, determined by the solution of the two PDE:
D[U[x,y,z,t],z]-D[S[x,y,t],t]-D[U[x,y,z,t},x]*D[S[x,y,t],x]-
D[u[x,y,z,t],y]*D[S[x,y,t],y]=0 at z= S[x,y,t]
Thanks. F Cap