A Comprehensive Differential Algebraic Framework for Dissipative Structures: Rigorous Theory, Explicit Solutions, and Certified Computations

A Comprehensive Differential Algebraic Framework for Dissipative Structures: Rigorous Theory, Explicit Solutions, and Certified Computations | 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 Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed.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. 31 October 2025 V1 Latest version Share on A Comprehensive Differential Algebraic Framework for Dissipative Structures: Rigorous Theory, Explicit Solutions, and Certified Computations Author : shifa liu 0009-0003-6570-2812 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.176194046.69921656/v1 183 views 138 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract 本文开发了一个全面且数学严谨的微分代数框架，用于构建耗散结构方程的显式解，建立和扩展代数几何和微分几何的方法。我们通过精心构建的递归附属过程提供了耗散结构微分闭包 KDiss 的完整定义，这些过程系统地结合了稳态、非线性反应、扩散算子、熵产生项和随机效应。在这个闭包中，我们证明了各种反应扩散系统、模式形成模型和非平衡热力学系统的显式解表示，并完全关注泛函分析设置、规律性条件和稳定性约束。该框架严格解决非线性、多尺度挑战和热力学兼容性，同时保持几何结构和物理一致性。我们提供详尽的建设性证明和详细的数学推导，推导出具有认证误差边界的显式解公式，并在适当的 Sobolev 和 Hölder 空间中建立精确的收敛标准。提出了具有精确复杂性分析的综合算法，包括稳定性保证、自适应精度控制和严格的验证协议。采用区间算术、结构保留离散化和分析解决方案的交叉验证的全面数值验证框架展示了在生物、化学和物理应用中的实际有效性。这项工作确定了耗散结构方程的显式解析解存在于适当构造的微分代数闭包中，为自组织提供了新的代数视角，同时保持与经典热力学理论和现代动力系统理论的完全一致性。对随机系统、高维问题、生物学应用和计算实现的扩展在数学和科学学科之间建立了深厚的联系，同时始终保持数学的严谨性。 Supplementary Material File (dissipative_structure_theory.pdf) Download 481.69 KB Information & Authors Information Version history V1 Version 1 31 October 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords certified computation differential algebraic closure dissipative structures entropy production nonlinear dynamics pattern formation reaction-diffusion equations self-organization stochastic pdes turing patterns Authors Affiliations shifa liu 0009-0003-6570-2812 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 183 views 138 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation shifa liu. A Comprehensive Differential Algebraic Framework for Dissipative Structures: Rigorous Theory, Explicit Solutions, and Certified Computations. Authorea . 31 October 2025. DOI: https://doi.org/10.22541/au.176194046.69921656/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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