An Integration--Annihilator method for analytical solutions of Partial Differential Equations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912380980559872 |
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| author | Richters, Oliver Glötzl, Erhard |
| author_facet | Richters, Oliver Glötzl, Erhard |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_11929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Integration--Annihilator method for analytical solutions of Partial Differential Equations Richters, Oliver Glötzl, Erhard Analysis of PDEs Mathematical Physics 35E20 (Primary) 35J05, 35Q99, 31B99, 35L05 (Secondary) G.1.8 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. |
| title | An Integration--Annihilator method for analytical solutions of Partial Differential Equations |
| topic | Analysis of PDEs Mathematical Physics 35E20 (Primary) 35J05, 35Q99, 31B99, 35L05 (Secondary) G.1.8 |
| url | https://arxiv.org/abs/2505.11929 |