{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/an-integration-annihilator-method-for","title":"An Integration--Annihilator method for analytical solutions of Partial Differential Equations","arxiv_id":"2505.11929","date":"2025-05-17","proceeding":null,"authors":["Oliver Richters","Erhard Glötzl"],"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.","url_abs":"https://arxiv.org/abs/2505.11929v1","url_pdf":"https://arxiv.org/pdf/2505.11929v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"an-integration-annihilator-method-for","repo_url":"https://zenodo.org/record/15363375","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}