Constructive Differential Algebraic Framework for Explicit Analytic Solutions to Nonlinear Partial Differential Equations

Constructive Differential Algebraic Framework for Explicit Analytic Solutions to Nonlinear Partial Differential Equations | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 14 October 2025 V1 Latest version Share on Constructive Differential Algebraic Framework for Explicit Analytic Solutions to Nonlinear Partial Differential Equations 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.176046789.94950004/v1 179 views 138 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文建立了一个全面的建设性微分代数框架，用于获得大类非线性偏微分方程（PDE）的显式解析解。我们定义并构建了非线性偏微分代数闭包 KNLPDE，这是一种通过递归附加过程构建的微分闭域扩展，该过程结合了线性偏微分方程、多索引根扩展、单位根和特定方程非线性特殊函数的解。在这个闭包中，我们证明了具有解析系数的 n 阶非线性偏微分方程的解，满足柯西-科瓦列夫斯卡娅条件，允许统一表示 u（x） = u0（x） + M −1 m=0 （Φm（c， x）） 1/pm ω km pm ψm（x）。该框架通过多参数同伦延续、自适应非线性基构建和多指数组合校正，严格解决偏微分方程固有的多维和无限维挑战。我们提供完整的建设性证明，推导非线性校正系数的显式组合表达式，建立收敛标准，并通过复杂度分析开发高效的计算算法。该框架在六个基本方程类别上进行了演示：Korteweg-de Vries、非线性薛定谔、爱因斯坦场方程、Monge-Ampère、正弦-戈登和反应-扩散系统，并进行了严格的数值验证，显示残差低于 10 −12 和频谱收敛率。这项工作确定，虽然对于许多非线性偏微分方程来说，初等函数中的闭式解是不可能的，但显式解析解存在于适当扩展和建设性定义的非线性偏微分代数闭包 KNLPDE 中。 Supplementary Material File (kdv.pdf) Download 433.05 KB Information & Authors Information Version history V1 Version 1 14 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords certified computation combinatorial analysis differential algebraic closure explicit solution homotopy continuation nonlinear 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 179 views 138 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Constructive Differential Algebraic Framework for Explicit Analytic Solutions to Nonlinear Partial Differential Equations. Authorea . 14 October 2025. DOI: https://doi.org/10.22541/au.176046789.94950004/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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