An Integration--Annihilator method for analytical solutions of Partial Differential Equations

We present a novel method to derive particular solutions for partial differential equations of the form $(\mathrm{A}+\mathrm{B}{)}^{k}Q(x)=q(x)$, with $\mathrm{A}$ and $\mathrm{B}$ being linear differential operators with constant coefficients, $k$ an integer, and $Q$ and $q$ sufficiently smooth functions. The approach requires that a function $W$ and an integer $\lambda$ can be found with the following two conditions: $q$ can be integrated with respect to $\mathrm{A}$ such that ${\mathrm{A}}^{\lambda +k}W(x)=q(x)$, and ${\mathrm{B}}^{\lambda +1}$ annihilates $W$ such that ${\mathrm{B}}^{\lambda +1}W(x)=0$. Applications include the Poisson equation $\mathrm{\Delta }Q(x)=q(x)$, the inhomogeneous polyharmonic equation ${\mathrm{\Delta }}^{k}Q(x)=q(x)$, the Helmholtz equation $(\mathrm{\Delta }+\nu )Q(x)=q(x)$ and the wave equation $\mathrm{\square }Q(x)=q(x)$. We show how solving the Poisson equation allows to derive the Helmholtz decomposition that splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field.