Solving a Nonlinear PDE in Maple 18 using Numeric Methods

[MathematicalPhysicist]

I wrote the next code:

restart;
pde := diff(u(x, t), t)+diff(u(x, t), x)-(diff(diff(u(x, t), x), t))^2 = u(x, t);
    
tmax := 0.5e-1; 
xmin := 0; 
xmax := 1;
 N := 10;
 bc1 := du(xmin, t) = 0;
 bc2 := u(xmax, t) = 0;
 ic1 := u(x, 0) = 1; 
ic2 := du(x, 0) = 2;
 bcs := {u(x, 0) = rhs(ic1), (D[1](u))(0, t) = rhs(bc1)};

pds := pdsolve(pde, bcs, numeric, time = t, range = 0 .. xmax, indepvars = [x, t], spacestep = (1/1000)*xmax, timestep = (1/1000)*tmax);
 
pds:-plot3d(t = 0 .. tmax, x = xmin .. xmax, axes = boxed, orientation = [-150, 80], shading = zhue, transparency = .1)

And I get the next two errors:

Error, (in pdsolve/numeric/plot3d) unable to compute solution for t>HFloat(0.0):
Newton iteration is not converging

Is there a method that will converge for this PDE?

Maybe Runga Kutta?

Thanks in advance.

Btw, the PDE that I had in mind is: $$u_x+u_t - (u_{xt})^2 = u$$ a nonlinear pde.

 

[Greg Bernhardt]

I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?