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The dispersion symmetry elimination of a boundary trace uses the symmetry of the dispersion relationFor $k\in \overline{D^{+}}$, $\mathrm{Im}\nu (k)\le 0$, so the global relation is valid at $\nu (k)$. Since $i\nu +\alpha =-ik$, it givesSubstitution into the contour integral representation produces an unwanted integral ${\int }_{\partial D^{+}}{e}^{ikx}\hat{u}(\nu (k),t)dk$. Its integrand is analytic in $D^{+}$ and decays on closing the contour upwards; Jordan lemma makes this integral zero for $x>0$. Hence the Fokas method eliminates the unknown normal derivative:All quantities here are determined by the prescribed initial and Dirichlet boundary data up to time $t$.

For verification and for numerical evaluation it is useful to evaluate the spectral contour integrals, giving a half-line drift reflection kernel. Putand define the half-line drift boundary kernelFubini's theorem, the Gaussian Fourier transform and contour deformation give the equivalent causal formulaFor the reflected initial term, ${\hat{u}}_0(i\alpha -k)={\int }_0^{\infty }{e}^{\alpha y+iky}{u}_0(y)dy$ on the contour; the evaluated Gaussian supplies the necessary large-$y$ decay. If $e^{\alpha y}{u}_0(y)$ is not integrable, truncate the initial conditions first, evaluate, and pass to the limit using Gaussian bounds. No extra exponential-decay assumption on the original data is needed for this kernel formula.

The boundary kernel follows particularly simply fromThis calculation independently checks both the sign and the coefficient of the boundary forcing.