Boundary value problem

A boundary value problem asks for a differential-equation solution satisfying conditions at more than one point or boundary component.

Solutions depend continuously on their forcing when convergence of the forcing in a specified norm implies convergence of solutions in a specified solution norm, with boundary conditions fixed. A uniform estimate $\Vert u-v{\Vert }_X\le C\Vert f-g{\Vert }_Y$ is a stronger quantitative form, namely Lipschitz continuity of the solution map. The monotonicity energy estimate for a semilinear Dirichlet problem is an example.

A boundary value problem with prescribed outward normal derivative, or the associated weighted flux, on the boundary. Its boundary data are Neumann boundary data. For an elliptic divergence-form operator without a zeroth-order term, integration over the domain imposes a compatibility condition between total source and boundary flux. A connected domain generally leaves a constant undetermined, fixed by a mean or reference value.

For a smooth bounded region $\Omega$, prescribing the outward normal derivative ${\partial }_{n}u=g$ of a harmonic function requiresThis follows from the divergence theorem applied to the gradient:On the unit disk, the condition becomes ${\int }_0^{2\pi }g(\theta )d\theta =0$. If $g={\sum }_{n\ge 1}(a_n\cos (n\theta )+b_n\sin (n\theta ))$ is sufficiently regular with zero mean, its regular harmonic expansion on a disk isThe factor $1/n$ compensates for the radial derivative. The arbitrary $C$ reflects the uniqueness of the Neumann problem up to an additive constant on a connected region.

On a connected bounded domain, two harmonic potentials with the same normal derivative differ by a constant. Their difference $u$ has ${\nabla }^{2}u=0$ and ${\partial }_{n}u=0$. Green's first identity gives $\int |\nabla u{|}^{2}dV={\int }_{\partial \Omega }u{\partial }_{n}udA=0$, so the gradient vanishes. In a multiply connected magnetic core, also fix the circulation around each independent hole: a curl-free magnetic field intensity with nonzero circulation need not be the gradient of a single-valued scalar potential. After those circulations have been matched, apply the scalar-potential uniqueness argument to the difference field.

For a sufficiently regular function on a domain $\Omega$, its boundary trace is the restriction to $\partial \Omega$, using the limit from within the domain when necessary. In a half-line evolution problem, the traces $f_j(t)={\partial }_x^{j}u(0,t)$ are boundary values of the spatial derivatives. An initial value does not generally determine every such trace without the partial differential equation and suitable boundary conditions. The Sobolev trace operator extends classical restriction continuously to appropriate Sobolev spaces, where pointwise values need not be defined.

An oblique derivative boundary condition prescribes a linear combination of the outward normal derivative and the tangential derivative on a boundary. Its strictly oblique case has a nonzero normal coefficient; a zero normal coefficient is a limiting tangential condition. Both trace directions must be tracked when constructing a global relation. On unbounded domains, regularity at corners and a growth condition at infinity are separate parts of an admissible boundary value problem.

For a solution with a continuous gradient at a quadrant corner, the prescribed derivative directions are $(-\sin {\beta }_1,\cos {\beta }_1)$ and $(\cos {\beta }_2,-\sin {\beta }_2)$. When ${\beta }_1+{\beta }_2=\pi /2$, these vectors are negatives of one another. Their corner data must therefore be opposite. Imposing $h_1(0)=h_2(0)$ as well forces both to vanish. Smooth decay of separate boundary functions alone does not ensure this local compatibility condition for an overdetermined linear system.

// Target: analysis.bigb

For a complex-valued harmonic function in the first quadrant, write $\Phi =q_z$ and $\Psi (\bar{z})=q_{\bar{z}}$ as independent holomorphic functions. Prescribe $i(e^{i{\beta }_1}\Phi -e^{-i{\beta }_1}\Psi )=g_1$ on the imaginary axis and $e^{-i{\beta }_2}\Phi +e^{i{\beta }_2}\Psi =g_2$ on the real axis. If ${\beta }_1+{\beta }_2=n\pi /2$, set $ϵ=(-1{)}^n$. Reflection into four quadrants produces a scalar Riemann-Hilbert problem with parity $H(-z)=-ϵH(z)$ and known jumps. Provided its corner and large-arc contributions vanish, the Cauchy integral formula givesComplex boundary data do not justify replacing $q_{\bar{z}}$ by the conjugate of $q_z$.

For real harmonic $q$, prescribe $-\sin {\beta }_1{q}_x+\cos {\beta }_1{q}_y=h_1$ on the vertical axis and $\cos {\beta }_2{q}_x-\sin {\beta }_2{q}_y=h_2$ on the horizontal axis. Their orthogonal derivatives $j_1,j_2$ determine the missing gradient components. If $H_1,J_1$ use the kernel $e^{ky}$ and $H_2,J_2$ use $e^{-ikx}$, the clockwise spectra ${\hat{q}}_1={\int }_0^{i\infty }{e}^{-ikz}{q}_{z}dz$, ${\hat{q}}_2=-{\int }_0^{\infty }{e}^{-ikz}{q}_{z}dz$ have the displayed expressions. Their global relation is ${\hat{q}}_1+{\hat{q}}_2=0$ in the third spectral quadrant.

// Target: analysis.bigb

Reality of the boundary derivatives gives $H_1(\bar{k})=\overline{H_1(k)}$ and $H_2(-\bar{k})=\overline{H_2(k)}$, and the same identities for $J_1,J_2$. The global relation and its conjugate express both quarter-plane oblique boundary spectral functions in terms of the known $H_1,H_2$ and the single bounded holomorphic function $\Phi$ in the upper half-plane. For $\gamma ={\beta }_1+{\beta }_2\in \{0,\pi /2,\pi \}$, its coefficients on the positive real and positive imaginary reconstruction rays are opposite, because $e^{4i\gamma }=1$. Closing the first-quadrant contour removes $\Phi$ by the Cauchy integral theorem.

// Target: analysis.bigb

For tangential data with ${\beta }_1={\beta }_2=0$, the harmonic function $q=xy$ has zero prescribed data on both axes but a growing nonzero derivative. For normal data with ${\beta }_1={\beta }_2=\pi /2$, $q=\log |z|$ also has zero prescribed data, and $q_z=1/(2z)$ decays at infinity but is singular at the corner. Thus decay of the data, and even decay of the solution gradient, does not alone determine the quadrant representation. Requiring the reflected gradient to decay at infinity and satisfy $|z|H(z)\to 0$ at the corner removes both examples. Without such conditions, add homogeneous holomorphic terms obeying the reflection parity.

Given a uniquely solvable Dirichlet problem, this map sends prescribed Dirichlet boundary data to the solution's outward normal derivative. It depends on the differential operator and domain. Spectral global relations offer one way to compute it numerically from boundary traces, while a finite collocation scheme still requires a rank and conditioning check.

In the semistrip Laplace spectral global relation, reality makes $W(\pm \kappa )$ real for $\kappa >0$. Its imaginary parts at those two arguments remove the unknown vertical-side normal derivative. With $G_j^s(\kappa )={\int }_0^{\infty }\sin (\kappa x)g_j(x)dx$ and $S_j(\kappa )={\int }_0^{\infty }\sin (\kappa x)q_y(x,j)dx$, the result isThe lower outward normal derivative is $-q_y$, so its Dirichlet-to-Neumann map coefficient is $-S_0$, whereas the upper coefficient is $S_l$.

// Target: analysis.bigb

Find functions analytic on the two sides of a contour whose boundary values obey a prescribed jump, with suitable normalization at infinity and other distinguished points. An additive jump $J$ on a counterclockwise closed contour is solved by the Cauchy integral formula $F(k)=(2\pi i{)}^{-1}\int J(\zeta )/(\zeta -k)d\zeta$, subject to the normalization and compatibility conditions. The Sokhotski–Plemelj theorem verifies the jump.

A Cauchy integral solves a prescribed boundary difference, not a prescribed sum. If $W_{+}+W_{-}=g$ on $(0,c)$, choose $\rho (z)=\sqrt{z(z-c)}\sim z$ at infinity. Its boundary values satisfy ${\rho }_{-}=-{\rho }_{+}$, so $V=W/\rho$ has difference $V_{+}-V_{-}=g/{\rho }_{+}$. Thus $W=\rho [C(g/{\rho }_{+})+H]$, where $C$ is the Cauchy integral with coefficient $1/(2\pi i)$ and $H$ has no jump. Endpoint integrability and normalization at infinity determine permissible regularizations and homogeneous terms. Simply using $C(g)$ for a sum jump is incorrect.

For the scalar jump $(a+b)w_{+}+(b-a)w_{-}=c$, divide by a nonvanishing canonical homogeneous solution $\tilde{w}$ to obtain an additive jump $\rho =c/((a+b){\tilde{w}}_{+})$. Its Cauchy integral gives a particular quotient, while every additional quotient $H$ with no jump is homogeneous freedom. The density $f=w_{+}-w_{-}$ therefore differs from the normalized particular density by $-2b{\tilde{w}}_{+}H/(a-b)$. Normalization, index and endpoint conditions determine the allowable $H$; dropping it without such conditions does not give the general solution.

Use the bounded columns supplied by Volterra analyticity sectors for reverse-dispersion mKdV and divide the necessary columns by their determinants. This gives a normalized meromorphic matrix on the six-ray contour $\arg k=m\pi /3$. The constant scattering data connection matrices generate its jumps by conjugation with $e^{i(kx-4k^{3}t){\sigma }_3}$. Zeros of column determinants require residue conditions; in a sufficiently small-data, zero-free regime only jumps are needed. The off-diagonal expansion coefficient reconstructs $q=-2i{\lim }_{k\to \infty }k{M}_{12}$ for this sign convention.

Combine columns of the three normalized Lax pair eigenfunctions in their common Volterra analyticity sectors for a half-line NLS Lax pair, dividing by the relevant determinants $a,d,d^{♯},a^{♯}$. The resulting matrix is normalized to the identity at infinity and has jumps on the real and imaginary axes. Connection matrices determine each jump explicitly. Zeros of those determinants require residue conditions or local contour deformations.

Suppose a scalar canonical factor has exterior growth $X_{-}(z)\sim z^q$ with index $-q<0$, and the original exterior unknown must decay as $O(z^{-1})$. Dividing by the canonical factor gives an additive jump $D_{+}-D_{-}=g$ with exterior decay $O(z^{-q-1})$. The exterior Cauchy integral expands in moments $-{\int }_L{t}^{j}g(t)dt/(2\pi i{z}^{j+1})$. Its first $q$ coefficients must therefore vanish, yielding the displayed solvability conditions. Under this decay there is no nonzero entire polynomial addition. A nonlocal moment appearing in $g$ must additionally agree with the moment of the reconstructed original jump.

Assume a regular invertible normalized Riemann-Hilbert problem solution $\mu =I+{\mu }_1/k+{\mu }_2/k^2+\cdots$ with jump tending to the identity and conjugated by $e^{-i(kx+2k^{2}t){\sigma }_3}$. Residuals of its two differential equations have no jump and vanish at infinity, so Liouville theorem gives a Lax pair. Coefficient comparison gives the displayed reconstruction and $A=2Q$; compatibility yields $i{Q}_t-Q_{xx}{\sigma }_3+2Q^3{\sigma }_3=0$. A jump matrix tending to zero instead is inconsistent with the normalization. Recovering the initial jump from a prescribed initial potential still requires direct scattering.

A rational jump with no zeros or poles on the contour can be split into factors analytic and nonvanishing on the two sides. The exterior factor may carry a power of $z$ recording the winding number. For a circle enclosing $0,\pm 1$, the jump $G=z/(z^2-1)$ has the canonical factors $X_{+}=1$, $X_{-}=z-z^{-1}$. The exterior growth of $X_{-}$ imposes compatibility conditions when a Cauchy transform is required to vanish at infinity.

The Wiener-Hopf method solves a boundary value problem with complementary half-line traces by Wiener-Hopf factorization and additive splitting into upper- and lower-half-plane analytic functions. Analytic continuation gives a common entire function, whose growth and edge conditions determine its admissible value.

In the upper half-plane, the displayed harmonic function has Dirichlet boundary condition value $x^{-1/4}$ on the positive axis, zero Neumann boundary condition value on the negative axis, and decay at infinity. Its negative trace is $\sqrt{2}|x{|}^{-1/4}$, but its positive normal derivative is $x^{-5/4}/4$. The latter requires a Hadamard finite-part integral in a Fourier transform. For the $e^{ikx}$ convention the transformed traces are $\sqrt{2}\Gamma (3/4)(ik{)}^{-3/4}$ below and $-\Gamma (3/4)(-ik{)}^{1/4}$ above. The homogeneous corner mode $r^{-1/2}\sin (\theta /2)$ produces the additional lower integral transform proportional to $k^{-1/2}$. Higher singular corner modes require a larger distribution class than a locally integrable negative trace.

For a scalar wave above a line with a Dirichlet half and a Neumann half, the boundary transform factors into ${\beta }_{+}(\alpha )=\sqrt{k+\alpha }$ and ${\beta }_{-}(\alpha )=\sqrt{k-\alpha }$. For $e^{-i\omega t}$, the limiting absorption principle puts $-k$ below and $k$ above the inversion contour. With Dirichlet conditions on the negative half-line and incident tangential wavenumber $a$, pole subtraction gives the diffracted trace $U_{+}=2i\sqrt{k-a}/[(\alpha +a)\sqrt{k+\alpha }]$. Its large-transform decay gives a bounded field with a square-root edge trace. The normal derivative has an inverse-square-root singularity on the Dirichlet half; it is identically zero on the Neumann half.

The plus factor is analytic and nonzero above the transform contour, and the minus factor below it. Their continuations inherit singularities and zeros in the opposite half-planes. Allocation of branch cuts and modal poles must agree with the physical outgoing continuation; no factor may have a zero inside its designated analytic domain.

A scalar Wiener-Hopf equation couples two Half-range Fourier transforms analytic in complementary half-planes. After dividing by a kernel factor, additive splitting separates the known forcing. Source poles, boundary terms and contour prescriptions are part of the problem data.

The equation ${\int }_0^{\infty }{e}^{-|x-t|}f(t)dt=f(x)$ has this solution family in the ordinary bounded-growth transform class. Differentiating twice gives $f^{\prime \prime }=-f$, and the integral's right derivative at zero gives $f^{\prime }(0)=f(0)$. Alternatively its positive-support Fourier transform is $A(ik-1)/(k^2-1)$; inversion above both real poles gives the same answer.

The scalar coefficient multiplying an unknown half-line transform in a Wiener-Hopf equation is its kernel. This is a transform-domain function, distinct from the null space of a linear map. Its analytic zeros, poles and winding number govern Wiener-Hopf factorization. For mixed scalar Helmholtz half-plane diffraction a typical kernel is the outgoing vertical wavenumber $\sqrt{k^2-{\alpha }^2}$.

For a nonvanishing kernel tending to the same nonzero value at both ends, the index counts its winding as the contour is traversed from left to right. Index zero permits a continuous logarithm tending to the same normalization at both ends. Canonical bounded Wiener-Hopf factorization and a constant common entire remainder require this qualification. For example $F(z)=(z+i)/(z-i)$ has index $-1$ on the real line; the bounded half-plane functions $w_{+}=(z-i)/(z+i)$ and $w_{-}=1$ obey $F{w}_{+}=w_{-}$. The analytic factors $F_{+}=z+i$, $F_{-}=z-i$ make their common product $z-i$, demonstrating why bounded input functions alone do not guarantee a constant factored remainder.

A forcing pole can be split between analytic half-planes by subtracting a factor's value at the pole. For instance,The second quotient has a removable pole and belongs to the plus expression; the first keeps the prescribed minus-side forcing pole. After Wiener-Hopf factorization, analytic continuation identifies an entire remainder, whose value is fixed by growth and edge conditions. The pole prescription must be stated together with the Fourier transform convention.

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.

For $q_t+q_x-q_{xxx}=0$ on $x>0$, its local relation has flux $(1+k^2)q-ik{q}_x-q_{xx}$. With $Q_0(k)={\int }_0^{\infty }{e}^{-ikx}q(x,0)dx$ and $G_j(k,T)={\int }_0^T{e}^{w(k)s}{\partial }_x^{j}q(0,s)ds$, its global relation is $e^{wT}\hat{q}=Q_0+(1+k^2)G_0-ik{G}_1-G_2$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}w(k)<0\}$, choose the other cubic root $\nu (k)$ in the lower half-plane. Then the known spectral density is $-Q_0(\nu )+(k^2-{\nu }^2)G_0-i(k-\nu )G_1$. The remaining final-state transform has zero integral around $\partial D_{+}$ by the Cauchy integral theorem and exponential decay. The full boundary includes the real axis and the hyperbola $\mathrm{Im}k=\sqrt{1+3(\mathrm{Re}k{)}^2}$, oriented with the domain on the left.

For a holomorphic function on $x>0$, $0<y<l$, define boundary spectra by counterclockwise traversal: ${\rho }_L={\int }_{il}^0{e}^{-ikw}f(w)dw$, ${\rho }_B={\int }_0^{\infty }{e}^{-ikx}f(x)dx$, and ${\rho }_T=-e^{kl}{\int }_0^{\infty }{e}^{-ikx}f(x+il)dx$. The displayed reconstruction follows from the Cauchy integral formula and the ray identity ${\int }_0^{e^{i\theta }\infty }{e}^{ik(z-w)}dk=i/(z-w)$ whenever that exponential decays. The three rays are selected by the inward normals of the three sides.

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For $\lambda >0$, put $a=k-\lambda /k$, $b=k+\lambda /k$. The spectral integral uses the positive real and positive imaginary rays, both oriented outward. On the first ray its density is ${\rho }_x=i{\int }_0^{\infty }{e}^{-iax}[bq(x,0)-q_y(x,0)]dx$; on the second it is ${\rho }_y={\int }_0^{\infty }{e}^{by}[aq(0,y)-i{q}_x(0,y)]dy$. The global relation ${\rho }_x+{\rho }_y=0$ holds in the third spectral quadrant. Primitives from the corner and from each infinite boundary end give an additive Riemann-Hilbert problem. Their limits are $-q$ at infinity and $q$ at zero, which explains the factor $1/(4\pi i)$.

The reflected global relation expresses the unknown value trace in terms of the prescribed derivative trace. Multiplying the half-line drift null contour identity by $(ik+\alpha )/(ik)$ removes the reflected time-dependent transform. Its pole at $k=0$ lies below the contour and is not crossed. The formula concerns the derivative $q_x(0,t)$, rather than the outward derivative $-q_x(0,t)$.

Let $I_C[F]=(2\pi {)}^{-1}{\int }_C{e}^{ikx-(k^2-i\alpha k)t}F(k)dk$. The drift-diffusion half-line global relation and the half-line drift null contour identity give the formula in the secondary title. Its real-space kernel is the difference of two drifted Gaussian heat kernels, and its boundary-forcing kernel is $x(2\sqrt{\pi }{t}^{3/2}{)}^{-1}\exp [-(x+\alpha t{)}^2/(4t)]$. This kernel is an approximate identity in time as $x\downarrow 0$, which verifies the Dirichlet boundary condition.

A spectral-parameter-dependent divergence identity equivalent to a partial differential equation. Integrating it over a space-time region gives a global relation between transformed initial and boundary data. For the half-line free Schrödinger equation, take $P=e^{-ikx+i{k}^{2}t}q$ and $Q=e^{-ikx+i{k}^{2}t}(-kq+i{q}_x)$.

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.

For the backward-sign Airy half-line global relation, suppose exactly one trace of derivative order $j\in \{0,1,2\}$ is unknown. Its spectral multiplier is a constant times $k^r$, $r=2-j$. Use the rotated null contours for half-range Fourier inversion with the displayed coefficients. The unknown contribution becomes an integral over the boundary of the middle upper sector $\pi /3<\arg k<2\pi /3$. There $\mathrm{Im}{k}^3<0$, sois bounded, while $e^{ikx}$ decays exponentially. The Cauchy integral theorem therefore removes that entire unknown contribution. The resulting solution depends on the initial transform and the two prescribed boundary transforms only.

// Target: analysis.bigb

For the linear dispersive Stokes equation, $\omega (\nu )=\omega (k)$ factors as $(\nu -k)({\nu }^2+k\nu +k^2-1)=0$. In $D_{+}=\{\mathrm{Im}k>0,\mathrm{Re}\omega (k)<0\}$ the two displayed other roots lie in the lower half-plane, so both can be used in the global relation. To check the root locations, at $k=ib$ they have imaginary part $-b/2$. A root could reach the real axis only where $\mathrm{Re}\omega (k)=0$, so continuity preserves their signs throughout this connected domain. The branch points $\pm 2/\sqrt{3}$ lie outside its closure. The weights $A_1=(k-{\nu }_2)/({\nu }_1-{\nu }_2)$ and $A_2=({\nu }_1-k)/({\nu }_1-{\nu }_2)$ obey $A_1+A_2=1$ and ${\nu }_j^{\prime }=-A_j$.

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.

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.

Let $F(k)={\int }_0^{\infty }{e}^{-ikx}{q}_0(x)dx$ and $G_0(k,t)={\int }_0^t{e}^{w(k)s}{g}_0(s)ds$. Using the roots of cubic dispersion symmetry with negative transport, put $\mathcal{T}F=[(k-{\nu }_2)F({\nu }_1)+({\nu }_1-k)F({\nu }_2)]/({\nu }_1-{\nu }_2)$. The Fokas method formula is an initial real-line integral minus an integral over $\partial D_{+}$ of $e^{ikx-wt}[\mathcal{T}F+(3k^2+1)G_0]$, both divided by $2\pi$. The boundary is oriented with $D_{+}$ on the left. Its unknown final-state transform has zero integral by analyticity and upper-half-plane decay.

For the Airy equation $q_t+q_{xxx}=0$ on $0<x<L$, define $F_j={\int }_0^t{e}^{-i{k}^{3}s}{\partial }_x^{j}q(0,s)ds$ and $G_j$ analogously at $L$. Set $\mathcal{F}=k^2{F}_0-ik{F}_1-F_2$ and $\mathcal{G}=k^2{G}_0-ik{G}_1-G_2$. Integrating the local relation with flux $-q_{xx}-ik{q}_x+k^{2}q$ proves the displayed global relation. The dispersion relation is invariant under multiplication of $k$ by a cube root of unity; the resulting three equations determine the three missing boundary transforms for data $q(0,t),q(L,t),q_x(L,t)$.

Here $\alpha =e^{2\pi i/3}$. The missing-trace matrix has determinant $-\sqrt{3}k\Delta (k)$. Its nonzero zeros are the eigenvalues in the parametrization $Av=i{k}^{3}v$ of $A=-d^3/d{x}^3$ with $v(0)=v(L)=v^{\prime }(L)=0$. Integration by parts gives $\mathrm{Re}⟨v,Av⟩=-|v^{\prime }(0){|}^2/2$. No nonzero zero lies in $\mathrm{Re}(-i{k}^3)\le 0$: on its boundary the determinant equation reduces, after rotation, to an equality between a unit-modulus exponential and $\cosh u-i\sqrt{3}\sinh u$, whose modulus exceeds one unless $u=0$. The zero at the origin is removable in the original boundary problem. There are nevertheless infinitely many genuine negative eigenvalues of $A$, obtained from $e^{-3s/2}=2\cos (\sqrt{3}s/2+\pi /3)$, $s>0$, with $k=-is/L$. Absence of contour poles in the Fokas method does not imply absence of discrete spectrum of the operator.

Let $q$ be a real harmonic function in a semistrip, $g_0(x)=q(x,0)$, $g_l(x)=q(x,l)$ and $h(y)=q(0,y)$. Let $G_j,N_j$ be the negative-exponential Half-range Fourier transforms of $g_j$ and $q_y(x,j)$, and put $H(k)={\int }_0^l{e}^{ky}h(y)dy$, $W(k)={\int }_0^l{e}^{ky}{q}_x(0,y)dy$. The closed differential form $e^{-ikz}{q}_{z}dz$ has zero boundary integral. Integration by parts of tangential derivatives cancels all corner values and gives the displayed global relation, for $\mathrm{Im}k\le 0$.

// Target: analysis.bigb

For $q_t=q_{xx}+\alpha q_x$ on $x>0$, write $Q(k)={\int }_0^{\infty }{e}^{-ikx}q(x,0)dx$ and $G_j(k,t)={\int }_0^t{e}^{\omega (k)s}{\partial }_x^{j}q(0,s)ds$. Twice integrating by parts gives $e^{\omega (k)t}\hat{q}(k,t)=Q(k)-G_1(k,t)-(ik+\alpha )G_0(k,t)$ in the lower transform half-plane. The dispersion relation is invariant under $k\mapsto i\alpha -k$. This global relation eliminates one unknown boundary trace after the symmetry maps the inversion contour into the transform's analytic domain.

For the linear dispersive Stokes equation $q_t+q_x+q_{xxx}=0$, let $\omega (k)=i(k-k^3)$. Its local relation is obtained with $X=q_{xx}+ik{q}_x+(1-k^2)q$. Integrating ${\partial }_t(e^{-ikx+\omega t}q)+{\partial }_x(e^{-ikx+\omega t}X)=0$ over the half-line gives the displayed global relation, with negative-exponential Half-range Fourier transforms and $G_j(k,t)={\int }_0^t{e}^{\omega (k)s}{\partial }_x^{j}q(0,s)ds$. The spatial transforms are holomorphic for $\mathrm{Im}k<0$, whereas the finite-time boundary transforms are entire functions.

Let $A_j$ be the weights of the cubic Stokes dispersion symmetry and set $F(k)={\sum }_j{A}_j(k){\hat{q}}_0({\nu }_j(k))$. Orient $\partial D_{+}$ with $D_{+}$ on its left. Evaluating the global relation at its two lower-half-plane roots and interpolating the linear expression $ik{G}_1+G_2$ removes both unknown boundary traces. The resulting Fokas method formula isThe discarded transform of the solution is holomorphic in $D_{+}$ and has zero integral after closing its contour for $x>0$. Initial compatibility and sufficient smoothness and spatial decay are required. At the boundary, use a fixed time horizon later than $t$ before Fourier inversion, to avoid an endpoint half-value.

Orient the upper-quadrant contour from $i\infty$ down to zero and then to $+\infty$. The reflected half-line transform has zero inverse contribution for positive $x$. Adding $c$ times this contribution, and deforming boundary integrands through the second quadrant where $|e^{ikx-i{k}^2(t-\tau )}|\le e^{-x\mathrm{Im}k}$, gives boundary coefficients $(1-c)k{\tilde{g}}_0-i(1+c){\tilde{g}}_1$. Thus $c=-1$ removes the unknown Neumann trace for a prescribed Dirichlet value, while $c=1$ removes the unknown Dirichlet trace for a prescribed Neumann value. The initial reflected term has coefficient $c$ and cannot be discarded after the time-dependent spectral decomposition.

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 $e^{-\omega (k)(t-s)}$.

In a Fokas method boundary integral over $\partial D_{+}$, where $\mathrm{Re}\omega <0$ inside, the contribution of replacing $G(k,t)$ by $G(k,T)$ is an integral of $e^{ikx}{\int }_t^T{e}^{\omega (k)(s-t)}g(s)ds$ times an holomorphic coefficient. For $x>0$ it vanishes by the Cauchy integral theorem and the exponential decay in the upper domain. The transformed amplitude is then independent of the observation time, which simplifies differentiation of the partial differential equation. Temporal inversion at $0<t<T$ recovers the full boundary value rather than the half-value of a truncated transform at its endpoint.

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 equal oriented Dirichlet boundary data on all three sides and a uniquely solvable elliptic problem, rotation invariance makes their outward normal derivatives equal. Pairing the global relation with its Schwarz conjugation of a spectral function at $\ell (k+\lambda /k)=2\pi in$ determines every Fourier coefficient of the common normal trace. The zero mode at $\lambda =0$ follows from zero total flux. At negative parameters, uniqueness and possible removable spectral denominators must be checked rather than assumed.