I need to numerically solve the sine-Gordon equation $$ \partial_{x,x} u(x,t) - \partial_{t,t} u(x,t) - \sin(u(x,t)) -\alpha \partial_t u(x,t) + \gamma =0 $$ for $x\in [0,15]$ and $t\in [0,2]$ with two boundary conditions $$\partial_x u(x,t)\vert_{x=0} = h, \hspace{0.5cm} \partial_x u(x,t)|_{x=15} = h + a_{ext} \sin(\omega_{ext} t). $$
My non-working Mathematica code is
al = 0.08; ga = 0.01; h = 5; aext = 3; omegaext = 1.4;
NDSolve[{D[u[x,t],x,x] - D[u[x,t],t,t] - Sin[u[x,t]] -al*D[u[x,t],t] + ga ==0,
(D[u[x, t],x]/. {x -> 0}) == h, (D[u[x, t], x] /. {x -> 15}) == h + aext Sin[omegaext t] }, u, {x,0,15},{t,0,2}]
I get an error message like this
NDSolve::fembdnl: The dependent variable in (u^(1,0))[0,t]==5 in the boundary condition DirichletCondition[(u^(1,0))[0,t]==5,x==0.] needs to be linear.
Can anyone suggest me how to fix this?