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

TL;DR

The paper introduces a new analytical method called the Integration--Annihilator method for finding particular solutions to certain classes of partial differential equations, simplifying solutions for equations like Poisson, Helmholtz, and wave equations.

Contribution

It presents a novel approach combining integration and annihilation techniques to derive solutions for PDEs with constant coefficient operators, expanding analytical solution methods.

Findings

Applicable to Poisson, Helmholtz, and wave equations

Enables derivation of Helmholtz decomposition from solutions

Provides explicit solution formulas for specific PDE classes

Abstract

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.