= Half-line Stokes solution with a boundary derivative
For the <linear dispersive Stokes equation> on $x>0$, with sufficiently smooth decaying initial data and $\operatorname{Re}p>0$, define $V(x,p)=\int_0^\infty R_p(x-y)u_0(y)\,dy$ and let $G(p)$ be the time <Laplace transform> of the prescribed derivative $u_x(0,t)$. Spatial decay and the one-dimensional decaying homogeneous space give
$$
U(x,p)=V(x,p)+\frac{e^{r_-x}}{r_-}\bigl[G(p)-V_x(0,p)\bigr].
$$
The <Neumann boundary condition> follows by differentiating at zero. The initial forcing follows from the <Stokes resolvent kernel with advection>. The <Bromwich inversion formula> gives an <integral representation> with no unknown boundary traces. Truncating or extending the prescribed boundary signal beyond a target time does not affect earlier values, by the <Laplace transform time-shift rule> and causality of this <Green function>.
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