http://mathematica.stackexchange.com/questions/80162/failing-to-solve-a-simple-pde-with-ndsolve#comment217289_80162

answer.nb

xmax = 3.;
zmin = 100.;
zmax = 121.;
tmax = 1.;
eqlist = {-170.2 (0.005 D[Ne[t, x, z], z] - D[Ne[t, x, z], {z, 2}]) - 
     3436. (0.005 D[Ne[t, x, z], x] - D[Ne[t, x, z], {x, 2}]) + 
     D[Ne[t, x, z], t] == 0., 
   DirichletCondition[Ne[t, x, z] == 1, z == zmin || z == zmax], 
   DirichletCondition[Ne[t, x, z] == 1, Abs[x] == 3], 
   Ne[0, x, z] == 1. + 10. E^(-99.9 x^2 - 1. (-110. + z)^2)};

sol2 = NDSolve[
   eqlist, {Ne}, {t, 0, tmax}, {x, -xmax, xmax}, {z, zmin, zmax}];