= Fourth-order Green kernel with two decay scales
{title2=$K(x)=\frac{e^{-a|x|}/a-e^{-b|x|}/b}{2(b^2-a^2)}$}
For $0<a<b$, the displayed decaying <Green function> solves $(-\partial_x^2+a^2)(-\partial_x^2+b^2)K=\delta_0$ on the real line. Its <Fourier transform> is
$$
\widetilde K(k)=\frac1{(k^2+a^2)(k^2+b^2)}
=\frac1{b^2-a^2}\left(\frac1{k^2+a^2}-\frac1{k^2+b^2}\right).
$$
Invert each term using the <one-dimensional modified Helmholtz Green function>. Then a decaying solution with forcing $h$ is the <convolution> $K*h$. For $a=1,b=2$, this gives $K=e^{-|x|}/6-e^{-2|x|}/12$ for the operator $\partial_x^4-5\partial_x^2+4$. Its first-derivative jump cancels, while its third-derivative jump is one, providing the unit <Dirac delta distribution> required by the fourth-order operator.
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