AF: Small: Algorithmic Algebraic Methods for Systems of Difference-Differential Equations

AF：小：差分微分方程组的算法代数方法

基本信息

批准号：
2139462
负责人：
Alexander Levin
金额：
$ 18.77万
依托单位：
Catholic University of America
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-05-01 至 2025-04-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2139462&HistoricalAwards=false
关键词：
AF Small Algorithmic Algebraic Methods

项目摘要

Differential, difference and difference-differential equations constitute main tools that scientists and engineers use to create mathematical models of real-life phenomena. Whereas continuous-time and discrete-time processes depending on several factors are described by systems of partial differential and difference equations, respectively, processes that include both continuous and discrete components (such as processes with time delay caused by the time required to transport mass, energy or information) are governed by systems of partial difference-differential equations (PDDEs). Furthermore, very often characteristics of physical, chemical or biological processes have certain symmetries, which can be captured mathematically as transformation group actions. Thus, the development of computational methods and algorithms for systems of PDDEs and such systems with group action is of primary importance in applications. Despite the over sixty-year history of constructive methods in differential and difference algebra, there are currently no efficient computational techniques for algebraic PDDEs. This project aims to develop the theory, methods and algorithms to determine the structure of solutions of systems of such equations including algebraic PDDEs with symmetry group actions. The research results will be applied to systems that describe mathematical models in physics, chemistry and biology. The educational goal of the project is to create an interdepartmental program on applications of symbolic computation that will involve undergraduate and graduate majors in computer science, mathematics, physics and biology at the Catholic University of America (CUA). The key research directions of this project are as follows. (1) Development of computational methods and algorithms for difference-differential elimination and for decomposition of solution sets of systems of algebraic PDDEs into unions of simple components. Extension of the obtained techniques to systems with group actions and/or weighted operators. (2) Development of algorithms for building Groebner-type bases in difference-differential modules and algebras. Applications of these algorithms to the computation of dimension functions of algebraic PDDEs that arise in applications. (3) Consistency analysis of finite difference approximations of algebraic differential equations via the techniques of generalized Groebner bases and difference-differential characteristic sets. (4) Application of the obtained methods and algorithms to systems of PDDEs that play fundamental roles in physics, engineering, chemical and biological modeling. The main methods and approaches of the project include the techniques of generalized difference-differential characteristic sets and relative Groebner bases, the use of dimension polynomials and quasi-polynomials, and decomposition methods for systems of algebraic PDDEs and such systems with group action and weighted basic operators. The results will be demonstrated in interdisciplinary research projects at CUA.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

差异，差异和差异方程式构成了科学家和工程师用来创建现实现象数学模型的主要工具。尽管持续时间和离散时间过程取决于多个因素，分别由部分差异方程和差异方程式的系统描述，这些过程包括包括连续和离散组件的过程（例如，由质量，能源或信息所需的时间延迟引起的时间延迟的过程）受部分差异方程式（PDDES）的系统控制。此外，通常的物理，化学或生物过程的特征通常具有某些对称性，可以在数学上以转化组的作用来捕获它们。因此，在应用程序中，针对PDDES系统和此类系统系统的计算方法和算法的开发在应用中至关重要。尽管在差异和差异代数方面有建设性方法的历史超过60年，但目前尚无代数PDDE的有效计算技术。该项目旨在开发理论，方法和算法来确定包括对称组动作的代数PDDES（包括代数PDDE）的系统解决方案的结构。研究结果将应用于描述物理，化学和生物学中数学模型的系统。该项目的教育目标是针对符号计算的应用创建一个部门间计划，该计划将涉及美国天主教大学（CUA）的计算机科学，数学，物理和生物学的本科和研究生专业。该项目的主要研究方向如下。（1）开发用于差异差消除的计算方法和算法，以及将代数PDDES系统的解分解为简单组件的工会。将获得的技术扩展到具有小组动作和/或加权操作员的系统。（2）开发用于在差异差异模块和代数中构建groebner型基础的算法。这些算法在应用中出现的代数PDDE的维数函数计算中的应用。（3）通过广义groebner碱基和差异差异特征集的技术对代数微分方程的有限差近似值的一致性分析。（4）将获得的方法和算法应用于在物理，工程，化学和生物学建模中扮演基本角色的PDDES系统。该项目的主要方法和方法包括广义差异差异特征集和相对groebner碱基的技术，尺寸多项式和准多项式的使用以及代数PDDES系统的分解方法以及具有团体动作和加权基础运营商的系统系统。该结果将在CUA的跨学科研究项目中证明。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。