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

TL;DR

The authors introduce an integration–annihilator method to construct analytic particular solutions for PDEs of the form $(A+B)^k Q=q$, where $A$ and $B$ are linear constant-coefficient operators. By identifying a function $W$ and an integer $bbambda$ that satisfy $A^{bbambda+k}W=q$ and $B^{bbambda+1}W=0$, they derive a closed-form $Q$ via a weighted sum of $A$ and $B$ acting on $W$, with a proof by induction. The framework is demonstrated on Poisson, polyharmonic, generalized Helmholtz, and wave-type equations, and is leveraged to streamline Helmholtz decompositions through Poisson solving. Compared to Green’s-function approaches, the method emphasizes symbolic, algebraic construction of particular solutions and includes a practical Mathematica worksheet. This yields new analytic avenues for solving linear PDEs and for deriving decomposition and potential-based representations in mathematical physics.

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 $λ$ can be found with the following two conditions: $q$ can be integrated with respect to $\operatorname{A}$ such that $\operatorname{A}^{λ+ k} W(x) = q(x)$, and $\operatorname{B}^{λ+ 1}$ annihilates $W$ such that $\operatorname{B}^{λ+ 1} W(x) = 0$. Applications include the Poisson equation $ΔQ(x) = q(x)$, the inhomogeneous polyharmonic equation $Δ^k Q(x) = q(x)$, the Helmholtz equation $(Δ+ ν) Q(x) = q(x)$ 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.

Figures (2)

• Figure 1: We suggest the following strategy to derive a particular solution for $\tilde{\Delta}^k Q(\bm x) = q(\bm x)$, the polyharmonic equation with a weighted, generalized Laplace operator $\mathop{\mathrm{D}}\nolimits = \tilde{\Delta} = \sum_{i=1}^n \omega_i \partial_{x_i}^2$ (the unweighted special case for $\mathop{\mathrm{D}}\nolimits = \Delta$ is the polyharmonic equation, and for $k=1$ is Poisson's equation): (1) expand $q$ into a sum of expressions $S(\bm x)$, (2) use the flowchart to find the appropriate solution for each expression $S(\bm x)$ separately, (3) sum all the resulting functions $Q_S(\bm x)$. In the diagram, $f(x_m)$ is any function of its argument. In Q1, if $S$ is a monomial and all $m$ satisfy the condition, we recommend to choose $m$ such that $x_m$ has the highest exponent to make $\lambda$ as small as possible and to have the fewest terms in the sum.
• Figure 2: This figure depicts how to obtain a Helmholtz decomposition for a vector field $\bm f$ by calculating the gradient field and the rotation field from the potential matrix $\mathbf{F} = \llbracket F_{ij} \rrbracket$ (dark blue short-dashed arrow). The variables and operators are defined in Definitions \ref{['def_helmholtz']}--\ref{['def_potentialmatrix']}. This paper derives methods to solve Poisson's equation (red long-dashed arrows) which allows to determine the potential vector $\phi$ from which all other quantities can be derived. In the traditional approach, often only in 3-dimensional space, Poisson's equation on unbounded domains only for sufficiently fast decaying fields for the source and rotation density, yielding a vector and scalar potential using the convolution of the Jacobian $\mathbf{J}$ with a solution $K$ of Laplace's equation is numerically solved.

Theorems & Definitions (19)

• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Lemma 4.3
• proof
• Corollary 5.1
• proof
• Corollary 5.2
• proof