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

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

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.