Unified Analytic Solution of Partial Differential Equations in Differential Algebraic Closure

Unified Analytic Solution of Partial Differential Equations in Differential Algebraic Closure | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 8 October 2025 V1 Latest version Share on Unified Analytic Solution of Partial Differential Equations in Differential Algebraic Closure Authors : Dongqi Liu 0009-0006-4018-9292 and shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175994542.29509652/v1 214 views 109 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文将微分代数闭包框架从常微分方程扩展到一类广泛的线性偏微分方程（PDE）。我们构建了一个严格定义的偏微分代数闭包（KPDE），并证明具有解析系数和非特征初始/边界条件的线性偏微分方程的解可以在该闭包中解析表达。解公式采用统一形式：u（x） = u0（x） + M −1 m=0 σm（Ψm（c， x）） 1/n ω mk n φm（x）， 0 ≤ k ≤ n − 1，其中 u0（x） 是特定解，c = （c1,. .., cM） 是初始/边界条件，Ψm ∈ KPDE 是显式偏微分多项式，包含从多元 Faà di Bruno 公式导出的组合校正项，ωn = e 2πi/n ，σm ∈ {+1， −1} 是由边界一致性确定的符号因子，φm（x） 是来自基本系统的基函数。我们提供了一个详细的建设性框架，推导出具有完整证明的校正组合表达式，提出了完整的算法描述，并将我们的结果置于经典偏微分方程理论和微分伽罗瓦理论中。数值实验证实了光谱收敛，并证明了对高阶方程进行组合校正的必要性。这项工作确定了对于一类重要的线性偏微分方程，显式解析解存在于适当构造的偏微分代数闭包KPDE中，为偏微分方程可解性提供了新的代数视角。 Supplementary Material File (unfied_pde.pdf) Download 441.10 KB Information & Authors Information Version history V1 Version 1 08 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords combinatorial analysis differential algebraic closure explicit solution numerical computation partial differential equations special functions Authors Affiliations Dongqi Liu 0009-0006-4018-9292 View all articles by this author shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 214 views 109 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Unified Analytic Solution of Partial Differential Equations in Differential Algebraic Closure. Authorea . 08 October 2025. DOI: https://doi.org/10.22541/au.175994542.29509652/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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