Neil, thanks for the advices. Somehow, it worked on my another PC.
I still have a question about your saying 'the term T^4 doesn't make sense to the PDE'. I don't understand how this works.
Here what I am trying to interpret is on the top surface, there is a radiation heat flow/loss (from the material to the environment), which is relevant to the material temperature.The other three boundaries don't have heat loss. But once I add it into my code, the result shows those errors again including 'the function T appears with no arguments'.
p = 2931.5;
c = 335;
k = 1.45;
\[Alpha] = 0.64;
\[Beta] = 1/(260*10^-6);
w = 0.0018;
L = 0.005;
tend = 10;
P = 20;
M = 0.005;
dd = NDSolve[{c*p*\!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]\(T[t, x, z]\)\) == k*(\!\(
\*SubscriptBox[\(\[PartialD]\), \({z, 2}\)]\(T[t, x, z]\)\) + \!\(
\*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\(T[t, x,
z]\)\)) + \[Beta]*\[Alpha]*(2*P)/(Pi*w^2)*
Exp[-((2*(x - 0.0025)^2)/w^2)]*Exp[-\[Beta]*z],
T[0, x, z] == 293.13, -
\!\(\*SuperscriptBox[\(T\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, 0] ==
0.64*5.67*10^-8*(293.13^4 - T^4),
\!\(\*SuperscriptBox[\(T\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "0", ",", "1"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, x, L] == 0,
\!\(\*SuperscriptBox[\(T\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, 0, z] == 0,
\!\(\*SuperscriptBox[\(T\),
TagBox[
RowBox[{"(",
RowBox[{"0", ",", "1", ",", "0"}], ")"}],
Derivative],
MultilineFunction->None]\)[t, M, z] == 0},
T[t, x, z], {t, 0, tend}, {z, 0, L}, {x, 0, M},
Method -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid"}}];
Plot3D[Evaluate[T[t, x, z] /. dd /. t -> 2], {x, 0, M}, {z, 0, L}]