# Fail to solve set of PDEs

I tried to solve a[t,z] and u[t,z] from set of PDES. Here a[t,z] describes laser incident on material where electrons osciallate at velocity u[t,z]. At boundary z=0, laser intensity is a[t,0]; at initial time t=0, electrons osciallate uniformally at velocity of u[0,z].

n=100;B=10;p=10^(-5);a0=0.01;u0=0.001
pdes={D[u[t,z],t]==-a[t,z]-p*n*T^(-3/2)*u[t,z]+I*u[t,z]*B, D[a[t,z],z]==-Im[k]*a[t,z]}/.k->(1-n/(1-B+I*p*n*T^(-3/2)))/.T->0.5*u[t,z]^2;
bc={a[t,0]==a0};
ic={u[0,z]==u0};
s=NDSolve[{pdes,bc,ic},{a,u},{t,0,1},{z,0,1}]
ContourPlot[Evaluate[Re[u[t, z]] /.s], {t, 0, 2}, {z, 0, 2}, PlotLegends -> Automatic]


The I got error message "NDSolve::femcnsd: The PDE coefficient -a[t,z]+10 I u[t,z]-(0.0149907 u[t,z])/(u[t,z]^(3/2))^(3/2)-(u^(1,0))[t,z] does not evaluate to a numeric scalar at the coordinate {1.,1.}; it evaluated to Indeterminate instead."

When I use T^(3/2) instead of T^(-3/2) and k instead of Im[k]. It works.

As suggested by user21, I used method in NDSolve

s = NDSolve[{pdes, bc, ic}, {a, u}, {t, 0, 2}, {z, 0, 2}, Method -> {"MethodOfLines"}]


Then I got error message which I do not understand: 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.

I still do not understand?

• Incorrect syntax. Use: T^(-3/2) not T^{-3/2} – Mariusz Iwaniuk Oct 7 '19 at 15:07
• Thank you. I corrected it. Same problems exist. – sixpenny Oct 7 '19 at 15:17
• Use the NDSolve method option Method -> {"MethodOfLines"}. That will give you a different set of messages and you need to address those issues; for example give an initial condition for a. – user21 Oct 7 '19 at 15:39
• Thank you. I tried and made progress. But still not solved, And I do not undertand the error message. – sixpenny Oct 7 '19 at 16:00
• What is the initial condition for a? – user21 Oct 8 '19 at 4:58