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MathGroup Archive 2004

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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


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