I am trying to solve a highly nonlinear system of coupled PDEs with NDSolveValue.

In the following you can find the system, the boundary conditions and the value of the numerical parameters.

and here is my code

```
Numerical parameters
e = 1.60217662*10^-19;
h = 6.62607004*10^-34;
(CurlyPhi)0 = h/(2*Pi*2*e);
(CapitalPhi)0 = h/(2*e);
Ic = 4*10^-6;
Lj0 = h/(2*Pi*2*e*Ic);
Cj = 32*10^-15;
Cgp = 64*10^-15;
Cgs = 62*10^-15;
L = 160*10^-12;
Ej0 = (h*Ic)/(2*e*2*Pi);
k = 0.9;
(CapitalPhi)dc = (2*Pi*(CurlyPhi)0)/3;
(Omega)p = 2*Pi*10^9*12;
(Omega)s = 2*Pi*10^9*8;
a = 30*10^-6;
s = NDSolveValue(
{
-Cj*(D(V(x, t) x, x, t, t) - Cgs/Cj*D(V(x, t), t, t))
+ (2*Ej0)/(CurlyPhi)0*(
Sin(1/(CurlyPhi)0*(D(V(x, t), x) + (1/2)*D(V(x, t), x, x)))*
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x)))) -
Sin(1/(CurlyPhi)0*(D(V(x, t), x) - (1/2)*D(V(x, t), x, x)))*
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x))))
) == 0,
Cgp*D(F(x, t), t, t) - 1/L*D(F(x, t), x, x) +
(k*Ej0)/(CurlyPhi)0*(
Cos(1/(CurlyPhi)0*(D(V(x, t), x) - (1/2)*D(V(x, t), x, x)))*
((1/
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x))))) Cos((CapitalPhi)dc/(
2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x)))*
Sin((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) - (1/2)*
D(F(x, t), x, x))))
- Cos(1/(CurlyPhi)0*(D(V(x, t), x) + (1/2)*D(V(x, t), x, x)))*
((1/
Abs(Cos((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x))))) Cos((CapitalPhi)dc/(
2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x)))*
Sin((CapitalPhi)dc/(2*(CurlyPhi)0) +
k/(2*(CurlyPhi)0)*(D(F(x, t), x) + (1/2)*
D(F(x, t), x, x))))
) == 0,
V(0, t) == ((CapitalPhi)0/20)*Sin((Omega)s*t),
V(x, 0) == 0,
F(0, t) == ((CapitalPhi)0/2)*Sin((Omega)p*t),
F(x, 0) == 0
},
{F, V},
{x, 0.003},
{t, 10^-9}
)
```

Now, every time I try to solve this system mathematica returns the following problem

Reading online I realized that this is a limit of mathematica 11.0 because my PDE contains a forth order derivative. For this reason I tried to change method of NDSolveValue using the following string

so the method of lines. But this time when I try to solve the system mathematica returns a different error message

This time I tried to change, add or remove some boundary conditions without really solving the problem.

Is there a different method I should try to solve this PDE system? Am I imposing wrong boundary conditions for the problem?

Thank you everybody for the help!