# Inconsistent boundary and initial conditions when solving sine Gordon equation

I have been trying to solve the following 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 $$t in (0,20)$$ with the boundary conditions

$$partial_x u(x,t)vert_{x=0} = h, hspace{2cm} partial_x u(x,t)vert_{x=15} = h + a_{mathrm{ext}} sin(omega_{mathrm{ext}}t),$$

and the initial conditions

$$partial_t u(x,t)vert_{t=0} = 0, hspace{2cm} u(x,t)vert_{t=0} = h x$$.

Here is my Mathematica code

``````const = {al -> 0.08, (Gamma) -> 0.01, h -> 6,  ax -> 2.5, (Omega)x -> 1.4};

NDSolveValue(({D(u(x, t), x, x) - D(u(x, t), t, t) - Sin(u(x, t)) - al D(u(x, t), t) + (Gamma) == 0,

(D(u(x, t), x) /. {x -> 0}) == h, (D(u(x, t), x) /. {x -> 15}) == h +  ax Sin((Omega)x  t),
u(x, 0) == h*x, (D(u(x, t), t) /. {t -> 0}) == 0} /.const), u, {x, 0, 15}, {t, 0, 20},

Method -> {"PDEDiscretization" -> {"MethodOfLines",
"SpatialDiscretization" -> {"TensorProductGrid",
"MinPoints" -> 200}}})
``````

The code works and gives a solution, but there appears an error message like this

``````NDSolveValue::ibcinc: Warning: boundary and initial conditions are inconsistent.
``````

From the second initial condition we have $$partial_x u(x,0) = h$$, which are consistent with the two boundary conditions at $$t=0$$. So I don’t understand why the above error occurs.

I have read many posts related to this problem. One of the solutions is to increase the “MinPoints” in the “Method”. I tried so, but it still did not not get rid of the error.

Any advice is very much appreciated.

Dat.

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