I want to solve the following PDEs numerically by NDSolve.

```
pde = {D((Alpha)((Tau), X, Y), (Tau)) -
I*(Kappa)(X, Y)*(Alpha)((Tau), X, Y) ==
(D((Alpha)((Tau), X, Y), X, X) +
D((Alpha)((Tau), X, Y), Y, Y)) + (1 - (Beta)((Tau), X, Y) -
Abs((Alpha)((Tau), X, Y))^2)*
(Alpha)((Tau), X, Y),
Subscript((Tau), o)*D((Beta)((Tau), X, Y), (Tau)) ==
(Subscript(D, d)/(Xi)^2)*(D((Beta)((Tau), X, Y), X, X) +
D((Beta)((Tau), X, Y), Y, Y)) +
(Subscript((CurlyPhi), o)^2/(Subscript(T, c)*
Subscript((Rho), n)*Subscript(C, v)))*
Grad((Kappa)(X, Y), {X, Y}) . Grad((Kappa)(X, Y), {X, Y}) -
((Beta)((Tau), X, Y) - Subscript(T, b)/Subscript(T, c))/
Subscript((Tau), eph),
Laplacian((Kappa)(X, Y), {X,
Y}) == (Subscript((Psi), o)^2/(4*e*Subscript(k, B)*
Subscript(T, c)*
Subscript((CurlyPhi), o)^2))*
Div(Im(ConjugateTranspose((Alpha)((Tau), X, Y))*
Grad((Alpha)((Tau), X, Y), {X, Y})),
{X, Y})}
bc = {(Beta)(0, X, Y) == Subscript(T, b)/Subscript(T, c),
(Beta)(0, 0, 0) == (Subscript(T, b) + dT)/Subscript(T, c),
(Beta)((Tau), l/2, Y) ==
Subscript(T, b)/Subscript(T, c), (Beta)((Tau), -(l/2), Y) ==
Subscript(T, b)/Subscript(T, c),
D((Beta)((Tau), X, Y), Y) == 0 /. Y -> w/2,
D((Beta)((Tau), X, Y), Y) == 0 /.
Y -> -(w/2), (Alpha)(0, X, Y) == 0.68, (Alpha)((Tau), l/2, Y) ==
0,
(Alpha)((Tau), -(l/2), Y) == 0,
D((Alpha)((Tau), X, Y), Y) == 0 /. Y -> w/2,
D((Alpha)((Tau), X, Y), Y) == 0 /. Y -> -(w/2),
D((Kappa)(X, Y), X) == ((-Subscript((Rho), n))*Is)/(w*d) /.
X -> l/2,
D((Kappa)(X, Y), X) == ((-Subscript((Rho), n))*Is)/(w*d) /.
X -> -(l/2),
D((Kappa)(X, Y), Y) == 0 /. Y -> w/2,
D((Kappa)(X, Y), Y) == 0 /. Y -> -(w/2)}
```

Here 𝛼 and 𝛽 are functions of 𝜏, X, Y, and 𝜅 is the function of X, Y.

However, the result shows there are inputs in error. (I use Mathematica 9.0)

Is that any problem in the setting of boundary conditions and initial values?

What is the problem?

The related parameters are:

```
Subscript(n, o) = 100;
Subscript((Rho), n) = 1;
Is = 10;
l = 80;
w = 20;
d = 5;
dT = 3;
Subscript(T, b) = 7.5;
Subscript(T, c) = 15;
Subscript((Tau), eph) = 0.0001;
Subscript((Tau), o) = 0.01;
e = 1;
Subscript(k, B) = 1;
Subscript((CurlyPhi), o) = 1;
Subscript((Psi), o) = 1;
Subscript(C, v) = 5;
(Xi) = 4.5;
Subscript(D, d) = 0.5;
```

or