Modified Helmholtz equation ($(\Delta -m^2)u=0$)

The modified Helmholtz equation isIt reduces to the Laplace equation when $m=0$.

Let $\mu >0$ and let $u$ be a real smooth solution on a bounded regular domain with zero Dirichlet boundary conditions. Green's first identity givesThe two sides have opposite signs unless both vanish, so $u=0$. This energy method also proves uniqueness for prescribed boundary data by applying it to the difference of two solutions. For complex solutions use the complex conjugate in the integral. This argument does not apply with the same sign to the Helmholtz equation, which can have nonzero zero-boundary eigenfunctions.

For $m,\beta >0$, set $K_m(r)=e^{-mr}/r$. Distributionally, $(\Delta -m^2)K_m=-4\pi \delta$, so the modified Helmholtz equation with unit delta source has Green function $-K_m/(4\pi )$. The difference $[K_{\beta }-K_m]/({\beta }^2-m^2)$ solves the equation with source $K_{\beta }$: the two delta terms cancel. It has no added point source at zero and decays at infinity. The Green function representation proves the displayed convolution with its positive sign.

At coincident rates, the continuous limit is $e^{-mr}/(2m)$; at $r=0$, it is $1/(m+\beta )$. The latter also follows by directly integrating $e^{-(m+\beta )r^{\prime }}d{r}^{\prime }$ after the spherical factors cancel. These limits prevent the formula's apparent singular denominators from being mistaken for physical divergences.

Here $B(p)=\sqrt{p^2+4\lambda }$ and $S_j$ are the Fourier sine transforms of the boundary values. The modified Helmholtz closed spectral one-form and its reflected global relations remove the unknown normal traces. Equivalently, four reflected fundamental solutions give a zero-boundary Green function in the quadrant. Its inward derivatives give the displayed formula. The two terms recover the bottom and left traces respectively; matching corner values ensure continuity at the corner.

For $k\ne 0$, differentiation gives $dW=-2e^{-ikz-\lambda \bar{z}/(ik)}(q_{z\bar{z}}-\lambda q)dz\wedge d\bar{z}$. Thus the differential form is closed precisely for the modified Helmholtz equation $\Delta q-4\lambda q=0$. Integrating around an oriented domain gives a global relation. The primitive $\mu e^{-ikz-\lambda \bar{z}/(ik)}$ also produces compatible first-order equations whose spectral jumps reconstruct $q$.

For $m>0$, the decaying fundamental solution of $-\Delta +m^2$ in two dimensions is $K_0(mr)/(2\pi )$, with $K_0$ the Modified Bessel function of the second kind. Green second identity reconstructs an interior solution from its Dirichlet boundary condition $f$ and outward normal derivative $h$ as $q(r)={\int }_{\partial \Omega }({\Gamma }_{m}h-f{\partial }_n{\Gamma }_m)ds$.

For $k>0$, a nonzero spectral parameter for a linear boundary value problem givesThe identity $A^2+B^2=k^2$ proves that this solves the modified Helmholtz equation. The reciprocal companion is $v_{1/\lambda }=\overline{v_{\overline{\lambda }}}$.

On a square, a normalized modified Helmholtz adjoint plane wave with positive normal growth toward one side has magnitude one there, magnitude $e^{-2\eta }$ on the opposite side, and an adjacent-side magnitude integral bounded by $1/\eta$. Thus large tangential spectral frequencies suppress remote-side normal-trace coefficients. Paired sine collocation of square modified Helmholtz global relations further cancels adjacent-side unknown traces exactly.

For a radial function in three dimensions, $(\Delta -1)y=g(x)$ becomes $y^{\prime \prime }+2y^{\prime }/x-y=g(x)$. Setting $u=xy$ gives $u^{\prime \prime }-u=xg(x)$, reducing it to a constant-coefficient linear ordinary differential equation. If $g=e^{-3x}/x^2$, a decaying particular solution obtained by variation of parameters isThe small-argument expansion of the exponential integral givesThe undetermined decaying homogeneous coefficient $\alpha$ is fixed by a boundary condition or by a matched asymptotic expansion.