Extension of the Differential Algebraic Framework to Nonlinear Partial Differential Equations: A Constructive Approach to Unified Analytic Solutions

Extension of the Differential Algebraic Framework to Nonlinear Partial Differential Equations: A Constructive Approach to Unified Analytic Solutions | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 13 October 2025 V1 Latest version Share on Extension of the Differential Algebraic Framework to Nonlinear Partial Differential Equations: A Constructive Approach to Unified Analytic Solutions 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.176037505.52454602/v1 150 views 132 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract This paper establishes a constructive differential algebraic framework for obtaining explicit ana lytic solutions to broad classes of nonlinear partial differential equations (PDEs). We define the non linear partial differential algebraic closure KNLPDE, a differentially closed field extension constructed through a recursive adjunction process that incorporates solutions to linearized PDEs, multi-index radical extensions, roots of unity, and a predefined set of nonlinear special functions. Within this closure, we prove that solutions to n-th order nonlinear PDEs with analytic coefficients, satisfying the conditions of the Cauchy-Kovalevskaya theorem, admit a unified representation. The framework rigorously addresses the multi-dimensional and infinite-dimensional challenges inherent in PDEs. We provide constructive proofs, derive explicit combinatorial expressions for nonlinear correction coeffi cients, and establish convergence criteria for the iterative construction of nonlinear basis functions. Detailed algorithms with complexity analysis are presented, including stability guarantees and adap tive precision control. A rigorous validation framework with certified error bounds is established, employing interval arithmetic and cross-verification against high-order numerical methods. This work demonstrates that while closed-form solutions in elementary functions are impossible for many nonlinear PDEs, explicit analytic solutions exist within the appropriately extended and construc tively defined nonlinear partial differential algebraic closure KNLPDE. The framework is shown to be consistent with classical PDE theory while extending the solution space to include nonlinear special functions and combinatorial correction structures. Supplementary Material File (nonlinear_pde3.pdf) Download 502.14 KB Information & Authors Information Version history V1 Version 1 13 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords adaptive precision certified computation combinatorial analysis differential algebraic closure explicit solution homotopy methods nonlinear partial differential equations numerical computation 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 150 views 132 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Dongqi Liu, shifa liu. Extension of the Differential Algebraic Framework to Nonlinear Partial Differential Equations: A Constructive Approach to Unified Analytic Solutions. Authorea . 13 October 2025. DOI: https://doi.org/10.22541/au.176037505.52454602/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 . Format Please select one from the list RIS (ProCite, Reference Manager) EndNote BibTex Medlars RefWorks Direct import Tips for downloading citations document.getElementById('citMgrHelpLink').addEventListener('click', function() { popupHelp(this.href); return false; }); $(".js__slcInclude").on("change", function(e){ if ($(this).val() == 'refworks') $('#direct').prop("checked", false); $('#direct').prop("disabled", ($(this).val() == 'refworks')); }); View Options View options PDF View PDF Figures Tables Media Share Share Share article link Copy Link Copied! Copying failed. 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