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Trouble solving system of first order, two variable PDEs in

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93341] Trouble solving system of first order, two variable PDEs in
  • From: Tim Mc <tmccomb at creol.ucf.edu>
  • Date: Tue, 4 Nov 2008 06:18:52 -0500 (EST)

Hello All,

I am trying to solve what will become a large system of PDEs (20-30 equations) (though now it is only three representative equations) using the Mathematica NDSolve function. The equations all only involve first derivatives and are a function of one time and one space variable. I have inserted the code below. When I attempt to evaluate I get the error:

NDSolve::ivone: Boundary values may only be specified for one \
independent variable. Initial values may only be specified at one \
value of the other independent variable

It seems no matter what I do (even stripping off the numbers and breaking the equations down to their bare bones) I always get this error for these type of boundary conditions. 

Can any one give me any advice on whether I am doing something wrong in the software or if my set of equations are simply incorrectly posed?

This is my Mathematica code:

\[Sigma]es = 3.4*^25;
\[Sigma]as = 5*^-27;
\[Sigma]ep = 5*^-26;
\[Sigma]ap = 8*^-25;
\[Tau] = 840*^-6;
\[Alpha] = 3*^-3;
Nn = 6*^25;
\[CapitalGamma] = 0.85;
\[CapitalGamma]p = 0.01;
\[Lambda]p = 0.915*^-6;
\[Lambda]s = 1.064*^-6;
A = 3*^-10;
h = 6.626*^-34;
c = 3*^8;
n = 1.5;

eq1 = D[N2[z, t], 
    t] == (\[CapitalGamma]p*\[Lambda]p/(h*c*A)) (\[Sigma]ap*(Nn - 
         N2[z, t]))*Pp[z, t] - 
    N2[z, t]/\[Tau] + (\[CapitalGamma]*\[Lambda]s/(h*c*
          A)) (\[Sigma]as*(Nn - N2[z, t]) - \[Sigma]es*N2[z, t])*
     Pp[z, t];

eq2 = D[Pp[z, t], z] + (n/c)*
     D[Pp[z, t], 
      z] == -\[CapitalGamma]p (\[Sigma]ap*(Nn - 
          N2[z, t]) - \[Sigma]ep*N2[z, t])*Pp[z, t] - \[Alpha]*
     Pp[z, t];

eq3 = D[Ps[z, t], z] + (n/c)*
     D[Ps[z, t], 
      z] == \[CapitalGamma] (\[Sigma]es*
        N2[z, t] - \[Sigma]as*(Nn - N2[z, t]))*Ps[z, t] - \[Alpha]*
     Ps[z, t];

bc1 = Ps[0, t] == 1;
bc2 = Pp[0, t] == 20;
bc3 = N2[z, 0] == 0;


solution = {N2, Pp, Ps} /. 
  First[NDSolve[{eq1, eq2, eq3, bc1, bc2, bc3}, {N2, Pp, Ps}, {z, 0, 
     10}, {t, 0, 0.001}]]


The error I see: 

NDSolve::ivone: Boundary values may only be specified for one \
independent variable. Initial values may only be specified at one \
value of the other independent variable. >>


Thank You in Advance,

Tim


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