Oliver Richters, Erhard Glötzl
An Integration–Annihilator method for analytical solutions ofPartial Differential Equations
O.R.: Potsdam Institute for Climate Impact Research, Potsdam, Germany.
E.G.: Institute of Physical Chemistry, Johannes KeplerUniversity Linz, Austria.
May 15, 2025–Version 1arXiv preprint arXiv:2505.11929
Abstract:We present a novel method to derive particular solutions for partial differentialequations of the form (A+B)kQ(x)=q(x), with A and B being linear differential operatorswith constant coefficients,kan integer, andQandqsufficiently smooth functions. Theapproach requires that a functionWand an integerλcan be found with the following twoconditions:qcan be integrated with respect to A such thatAλ+kW(x)=q(x), and Bλ+1annihilatesWsuch that Bλ+1W(x)=0.Applications include the Poisson equation∆Q(x)=q(x), the inhomogeneous polyharmonicequation∆kQ(x)=q(x), the Helmholtz equation (∆ +ν)Q(x)=q(x) and the wave equation□Q(x)=q(x).We show how solving the Poisson equation allows to derive the Helmholtzdecomposition that splits a sufficiently smooth vector field into a gradient field and adivergence-free rotation field.Keywords:Partial Differential Equations, Particular Solution, Annihilator Method, HelmholtzEquation, Poisson Equation, Polyharmonic Equation, Wave Equation, HelmholtzDecomposition