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Interpret the printed coordinate notation as the square corners $1+i$, $1-i$, $-1-i$, $-1+i$. This listed order is clockwise, contrary to the counterclockwise convention in (ii). Keep the printed first side directed from top to bottom. Then $m_1=1$, $h_1=-i$, and $z(s)=1-is$. Put $g_1(s)=u(1,-s)$ and $q_1(s)=u_x(1,-s)$, where $q_1$ is the outward normal derivative on the right side. The pullback formula, without any orientation shortcut, givesIndeed $h_1{u}_z-{\bar{h}}_1{u}_{\bar{z}}=-i(u_z+u_{\bar{z}})=-i{u}_x$, and $i\beta (\lambda h_1+{\bar{h}}_1/\lambda )=\beta (\lambda -{\lambda }^{-1})$. If one reverses the side to match the counterclockwise convention, its parameter is $1+is$ and its integrand is $-W_1(-s,\lambda )$. Reversing every side multiplies the global relation by minus one, leaving its zero value unchanged.

The unheaded numerical reconstruction request. The four unknown Neumann boundary conditions are four functions on the sides, not four scalar values. The polygonal global relation is linear in their outward normal derivatives, and its remaining terms depend only on the prescribed Dirichlet boundary data.

Choose a finite approximation on each side, for example an expansion of $q_j(s)$ in Legendre polynomials or piecewise polynomials. Substitute those expansions into the consistently oriented global relation. At chosen nonzero spectral parameters for a linear boundary value problem, integrate the exponential kernels against each basis function to assemble a complex linear system; the known right-hand side is obtained by integrating the given Dirichlet boundary data. Use enough independent samples to resolve all side coefficients, and preferably oversample. The conjugate global relations for the modified Helmholtz equation provide useful companion tests; for complex data use the corresponding independent adjoint relation rather than assuming the traces real.

Solve the scaled system by least-squares solution using a stable factorization such as a singular value decomposition. Sampling directions should probe all sides, and exponential row scaling avoids overflow and poor conditioning. Refine the side approximation and spectral samples until the recovered traces and unused global relation residuals stabilize. Corner incompatibilities or limited corner regularity call for mesh refinement or enriched endpoint basis functions. This realizes a numerical Dirichlet-to-Neumann map without first discretizing the whole interior.

The underlying Dirichlet problem is uniquely solvable in the usual trace class for $\Delta u-4{\beta }^{2}u=0$ with $\beta >0$: the homogeneous problem has ${\int }_{\Omega }(|\nabla u{|}^2+4{\beta }^2|u{|}^2)dA=0$. This supports the boundary reconstruction, although uniqueness of the continuous problem alone does not guarantee that an arbitrary finite set of spectral samples is well conditioned.

Fix the convention $\Delta q-m^{2}q=0$, $m>0$, and let $c_m=m^2/4$. With $z=x+iy$, two families of modified Helmholtz adjoint plane waves areBoth satisfy $\Delta E_{\pm }=m^2{E}_{\pm }$, since the products of their $z$ and $\bar{z}$ exponents are $m^2/4$. They obey $E_{-}(z,k)=\overline{E_{+}(z,\bar{k})}$. Green second identity gives the two conjugate global relations for the modified Helmholtz equationFor real boundary traces, ${\mathcal{G}}_{-}(k)=\overline{{\mathcal{G}}_{+}(\bar{k})}$. Thus the second identity is the conjugate spectral companion, rather than an unrelated extra boundary condition. In complex differential form the first relation is equivalentlyIts integrand is a closed one-form: differentiating its coefficients gives $2E_{+}(q_{z\bar{z}}-c_{m}q)=0$. This supplies a direct derivation of the global relation as well as its outward-normal sign convention.

Write the known Dirichlet boundary condition on side $j$ as $f_j(s)$ and the unknown outward normal derivative as $h_j(s)$, with $-1<s<1$. For any linear combination $v$ of the adjoint waves, the global relation isExpanding the $h_j$ in a finite basis and enforcing these identities at selected collocation points for a global relation gives a linear system with a known right side. The square permits especially simple paired tests.

Let $p_n=n\pi /2$, ${\eta }_n=\sqrt{m^2+p_n^2}$, and ${\phi }_n(s)=\sin [p_n(s+1)]$ for $n\ge 1$. Define four real adjoint solutionsThey satisfy the modified Helmholtz equation because ${\eta }_n^2-p_n^2=m^2$. Each is a linear combination of two adjoint exponentials, hence is obtained from the two spectral global relations. More explicitly, put ${\rho }_n^{\pm }=({\eta }_n\pm p_n)/2$, so ${\rho }_n^{+}{\rho }_n^{-}=c_m$. For the $E_{+}$ convention above, the required spectral pairs areAppropriate phase-weighted differences produce $\sin [p_n(s+1)]$. For example the top test is $e^{-{\eta }_n}[e^{i{p}_n}{E}_{+}({\rho }_n^{-})-e^{-i{p}_n}{E}_{+}({\rho }_n^{+})]/(2i)$. The conjugate relation ensures a real system for real data. This is sine collocation of square modified Helmholtz global relations.

The functions ${\phi }_n$ are an orthonormal Fourier sine basis on $[-1,1]$. Set $h_{j,n}={\int }_{-1}^1{h}_j{\phi }_{n}ds$ and compute the known quantity $R_{j,n}={\int }_{\partial \Omega }f{\partial }_n{v}_{j,n}ds$ by numerical integration. Since ${\phi }_n(\pm 1)=0$, each adjoint test vanishes on both adjacent sides. On its own side it equals ${\phi }_n$, and on the opposite side it equals ${\delta }_n{\phi }_n$, where ${\delta }_n=e^{-2{\eta }_n}$. Thus the square modified Helmholtz Dirichlet-to-Neumann coefficients obeyEach block is inverted explicitly: for opposite sides $j,j^{\prime }$,Compute these coefficients for $1\le n\le N$ and reconstruct $h_j^{(N)}={\sum }_{n=1}^N{h}_{j,n}{\phi }_n$. This is a semi-analytical scheme: the spectral basis integrals and two-by-two inverses are explicit, while the known boundary integrals are evaluated numerically. Increase $N$ and the quadrature resolution until the desired convergence is observed; compatible smooth side data have convergent normal-trace expansions, while corner singularities require the usual weaker trace interpretation and more careful quadrature.

The requested weak interaction between sides is particularly clear: the adjacent-side unknown traces contribute exactly zero, and the opposite-side coefficient is $e^{-2\sqrt{m^2+(n\pi /2{)}^2}}\to 0$. The own-side coefficient stays equal to one. Thus diagonal dominance of paired square global-relation collocation is genuine after pairing; individual uncombined exponential samples need not have the same conditioning. The eigenvalues of each block are $1\pm {\delta }_n$, so its condition number is $(1+{\delta }_n)/(1-{\delta }_n)$.

There is also a direct localization check for an individual adjoint wave. On side $j$ with outward unit normal $n_j$ and tangent $t_j$, the normalized test $e^{\eta (n_j\cdot r-1)+ip{t}_j\cdot r}$ has $\eta =\sqrt{m^2+p^2}$. Its opposite-side magnitude is $e^{-2\eta }$, and its magnitude integrates to at most $1/\eta$ along either adjacent side. This side localization of modified Helmholtz plane waves explains why suitable large spectral parameters suppress remote-side effects even before the exact sine cancellation.

Finally the interior solution can be evaluated from the computed traces using the two-dimensional modified Helmholtz fundamental solutionHere $K_0$ is the Modified Bessel function of the second kind, and $(-\Delta +m^2){\Gamma }_m=\delta$. Replace $h$ by its computed finite Fourier sine series and use numerical integration; at interior points the boundary kernels are smooth. The sign follows from Green second identity with this fundamental-solution convention. This completes the numerical integration of the boundary value problem with a Dirichlet boundary condition, rather than stopping at an equation for its unknown boundary derivatives.

Let $\Omega$ be a bounded piecewise smooth domain, $f=u{|}_{\partial \Omega }$ and $q={\partial }_{n}u$, with $n$ the outward normal vector. Integrating the divergence form from part (i) and using the divergence theorem gives two global relations for a linear boundary value problem:These conjugate global relations for the modified Helmholtz equation are identities between the prescribed Dirichlet boundary condition and its unknown normal derivative; neither is a pointwise boundary equation. Both hold for every nonzero boundary spectral parameter under the usual trace regularity assumptions.

If $u$ is real, its two boundary traces $f,q$ are real. Since ${\tilde{v}}_{\lambda }=\overline{v_{\overline{\lambda }}}$ and the outward normal vector is real,Thus apply complex conjugation to the first global relation for a linear boundary value problem, and replace its parameter by $\overline{\lambda }$, to get the second. For complex $u$, conjugation would also replace its traces by their conjugates, so that argument requires the stated reality assumption. There is also an algebraic redundancy in this particular parametrization: ${\tilde{v}}_{\lambda }=v_{1/\lambda }$, hence $\tilde{\mathcal{G}}(\lambda )=\mathcal{G}(1/\lambda )$ even for complex data. This does not change the requested conjugation identity.