A tricky PDE system

*To*: mathgroup at smc.vnet.net*Subject*: [mg46867] A tricky PDE system*From*: mikulamali at hotmail.com (Mikula Barnes)*Date*: Fri, 12 Mar 2004 02:02:52 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

I have the following PDE system to solve. The variables are vector R[z,t]={r_i[z,t]} subject to boundary condition R[z,t=0]=R0 scalar a[z,t] with boundary condition a[z=0,t]=f[t] scalar b[z,t] with boundary condition b[z=L,t]=g[t] (a and b are two laser pulses counter-propagating through a medium of length L and the density of the medium - atomic populations - is described by vector R, and i=1,..,4). The equations to be solved subject to the above b.c. are: d(r_i[z,t])/dt = Xi[ R[z,t], a[z,t], b[z,t] ] d(a[z,t])/dz = F[ R[z,t], b[z,t] ] d(b[z,t])/dz = G[ R[z,t], a[z,t] ] where Xi, F, and G are linear functions. I tried using NDSolve to solve the system but Mathematica doesn't really recognize it as a PDE system. And I'm note sure that method of lines is aplicable here since boundary conditions for fields a[z,t] and b[z,t] are on the different sides of the medium. I know that physically the problem is well-defined. Is there perhaps any form of a differential-algebraic system that one could do here to solve the system? Setup my own difference-approximation acheme? Any other ideas? Thanks, -mik