Papers › An Integration--Annihilator method for analytical solutions of Partial Differential Equations
An Integration--Annihilator method for analytical solutions of Partial Differential Equations
Oliver Richters, Erhard Glötzl
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We present a novel method to derive particular solutions for partial differential equations of the form (A + B)ᵏ Q(x) = q(x), with A and 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 λ can be found with the following two conditions: q can be integrated with respect to A such that A^(λ+ k) W(x) = q(x), and B^(λ+ 1) annihilates W such that B^(λ+ 1) W(x) = 0. Applications include the Poisson equation ΔQ(x) = q(x), the inhomogeneous polyharmonic equation Δᵏ Q(x) = q(x), the Helmholtz equation (Δ+ ν) Q(x) = q(x) and the wave equation 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.
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