       PDE with boundary condition ODE

• To: mathgroup at smc.vnet.net
• Subject: [mg50214] PDE with boundary condition ODE
• From: mathma18 at hotmail.com (Narasimham G.L.)
• Date: Sat, 21 Aug 2004 03:04:15 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```Beginning attempt to solve for z(x,y)(constant negative Gauss
curvature)in Monge form PDE r*t-s^2 == -(1+p^2+q^2)^2, with ODE BC as
p(x,0)^2=(1+z^2)/(2-z^2) for boundary y=0 has mixed lists. TIA for
help.

Clear[x,y,z,z2]; xm=.6; ym=3;
pde=-(1+D[z[x, y], x]^2+D[z[x, y], y]^2)^2 ==
(D[z[x, y],{x,2}]*D[z[x, y],{y,2}]-D[z[x, y],{x,y}]^2);
BC= {D[z[x, 0],x]^2 == (1+z[x,0]^2)/(2-z[x,0]^2),
z[0, 0]== .01,D[z[x,0],x]==.01};
"flat BC= {z[x,0]==x^2/5}" ;
NDSolve[{pde,BC},z,{x,0,xm},{y,0,ym}];
z2[u_,v_]=z[u,v]/.First[%];
Plot3D[z2[x,y],{x,0,xm},{y,0,ym}];

```

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