Fokas method

The Fokas method uses a global relation for a linear boundary value problem between transformed boundary traces, complex spectral symmetries, and contour integration to eliminate unknown traces or construct a numerical boundary scheme. The distinction between the domains of analyticity of the transforms and the decay sectors of spectral exponentials controls valid contour deformations.

To eliminate an unknown boundary transform by a dispersion relation symmetry, evaluate the global relation at $\nu (k)$ only where the spatial transform remains analytic. Substitute the resulting trace identity into the contour representation. An unwanted transform vanishes only if its transformed argument and exponential admit a valid contour deformation and decay estimate. For constant drift, $\nu (k)=i\alpha -k$ maps the upper domain $\mathrm{Re}(k^2-i\alpha k)<0$ into the lower transform half-plane.

If $P(k)$ is analytic in the upper half-plane with suitable growth, paired integrals along the positive real and imaginary rays cancel when their exponential is $e^{ik(x+iy)}$ with $x,y>0$. For $0<y<\ell$, the analogous pair along the imaginary and negative real rays cancels with $e^{ik(x+i(y-\ell ))}$. These two applications of the Cauchy integral theorem eliminate reflected unknown transforms from a strip global relation; an individual ray integral need not vanish.