Global relation for a linear boundary value problem ($\mathcal{G}(\lambda )=0$)

A global relation is an identity connecting transforms or integrals of the boundary traces of a solution. For a formally self-adjoint operator $L=\Delta -k^2$, Green second identity givesfor every homogeneous adjoint solution $Lv=0$. Exponential adjoint families turn this into a spectral identity between the Dirichlet boundary condition and unknown normal derivative.

A boundary trace transformed over a finite time interval using the dispersion relation of a linear partial differential equation. For integrable trace data and a polynomial dispersion function, it is an entire function of the complex spectral parameter. Its growth depends on $\mathrm{Re}\omega (k)$; multiplying by $e^{-\omega (k)t}$ changes the integrand to the causal factor $e^{-\omega (k)(t-s)}$.

For $u_{z\bar{z}}={\beta }^{2}u$, pull back its spectral closed differential one-form along each straight side $z=m_j+s{h}_j$. Under counterclockwise traversal and outward normal derivatives $q_j$, the side density is $i{e}^{-i\beta (\lambda z-\bar{z}/\lambda )}[|h_j|q_j+\beta (\lambda h_j+{\bar{h}}_j/\lambda )g_j]$, where $g_j$ is the Dirichlet boundary data. The Generalized Stokes theorem gives the displayed global relation. Clockwise traversal reverses the normal-derivative coefficient; all orientations must be changed consistently.

For the advection-diffusion equation $u_t=u_{xx}+\alpha u_x$ on $x>0$, the spectral divergence form uses $\omega =k^2-i\alpha k$ and factor $e^{-ikx+\omega t}$. With negative-exponential Half-range Fourier transforms, integration gives the displayed global relation for $\mathrm{Im}k\le 0$. The boundary transform $G_1$ refers to $u_x(0,t)$, the negative of the outward normal derivative.

These are chosen complex values of a spectral parameter for a linear boundary value problem at which a truncated global relation for a linear boundary value problem is enforced. Their directions and magnitudes determine which boundary side and which tangential mode are tested. Suitable row combinations can give much better conditioning than arbitrary raw samples.

The two modified Helmholtz adjoint plane waves $v_{\lambda }$ and ${\tilde{v}}_{\lambda }=\overline{v_{\overline{\lambda }}}$ give boundary identities $\mathcal{G}(\lambda )=0$ and $\tilde{\mathcal{G}}(\lambda )=0$. For real solution traces, $\tilde{\mathcal{G}}(\lambda )=\overline{\mathcal{G}(\overline{\lambda })}$. Reality is needed for this conjugation argument; the parametrization separately yields the reciprocal identity $\tilde{\mathcal{G}}(\lambda )=\mathcal{G}(1/\lambda )$.

On the square $[-1,1{]}^2$, write normal traces in ${\varphi }_n(s)=\sin [n\pi (s+1)/2]$. Set $p_n=n\pi /2$, ${\omega }_n=\sqrt{k^2+p_n^2}$, $r_n=({\omega }_n+p_n)/k$. Paired spectral samples $\{r_n,r_n^{-1}\}$, $\{-r_n,-r_n^{-1}\}$, $\{i{r}_n,i/r_n\}$, $\{-i{r}_n,-i/r_n\}$ give adjoint tests $e^{\mp {\omega }_{n}y}{\varphi }_n(x)$ and $e^{\mp {\omega }_{n}x}{\varphi }_n(y)$. Each test vanishes on the adjacent sides, removing their unknown normal derivatives.

Let $R_{jn}={\int }_{\partial \Omega }f{\partial }_n{v}_{jn}ds$ for a square sine adjoint test, and put ${\rho }_{jn}=e^{-{\omega }_n}{R}_{jn}$, ${\delta }_n=e^{-2{\omega }_n}$. The opposite-side normal-trace coefficients obeywith the same formulas for $L,R$. When the known Dirichlet boundary condition is integrated exactly, these are exact coefficients, since orthogonality eliminates all other unknown modes.

For sine collocation of square modified Helmholtz global relations, each opposite-side mode has a scaled matrixThe assembled system has strict diagonal dominance, and its spectral condition number of a positive-definite matrix is below $(1+e^{-\pi })/(1-e^{-\pi })<1.091$. The conclusion concerns the explicit paired sine rows, not arbitrary uncombined complex rows.