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 $(\operatorname{A} + \operatorname{B})^k Q(x) = q(x)$, with $\operatorname{A}$ and $\operatorname{B}$ 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 $\operatorname{A}$ such that $\operatorname{A}^{\lambda + k} W(x) = q(x)$, and $\operatorname{B}^{\lambda + 1}$ annihilates $W$ such that $\operatorname{B}^{\lambda + 1} W(x) = 0$. Applications include the Poisson equation $\Delta Q(x) = q(x)$, the inhomogeneous polyharmonic equation $\Delta^k Q(x) = q(x)$, the Helmholtz equation $(\Delta + \nu) Q(x) = q(x)$ and the wave equation $\Box Q(x) = q(x)$. 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.