AF: Small: Computational Methods for Difference-Differential Equations

AF：小：差分微分方程的计算方法

• 批准号: 1016608
• 负责人: Alexander Levin
• 金额: $ 14.31万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016608&HistoricalAwards=false
• 关键词: AF; Small; Computational; Methods; Difference

This research is aimed at developing constructive methods and algorithms for computational analysis of systems of partial algebraic difference-differential equations (PADDEs) and description of their solution sets. Such systems arise in a wide variety of problems in mathematics and its applications including mathematical physics, automatic control, dynamical systems, mechanics, molecular chemistry, and cellular biology.The key research objectives of this project are: (1) development of the theoretical foundation and algorithms for difference-differential elimination, in particular for decomposing solution sets of systems of PADDEs into unions of "simple" sets; (2) extension of the constructive methods of difference-differential algebra to the computational analysis of systems of partial differential equations (PDEs) with group action (this is of special interest for applications, since the solutions of fundamental systems of PDEs governing physical fields must be invariant with respect to certain group actions); (3) elaboration of methods and algorithms for computation of dimension polynomials that express Einstein's strength of a system of PADDEs. Such algorithms, in particular, will allow one to choose optimal (in the sense of A. Einstein) systems of PDEs for mathematical models of physical processes.The main methods and approaches to be used include the characteristic set technique, which will be extended to rings of difference-differential polynomials, generalized Groebner basis method for difference-differential modules, the technique of univariate and multivariate dimension polynomials, and decomposition methods for PADDEs.Despite the over sixty-year history of algorithmic approaches in differential and difference algebra, initiated by J. Ritt, E. Kolchin, R. Cohn and recently expanded by M. Bronstein, X. Gao, P. Hendrics, and M. Singer, among many others, there are no computational methods efficient enough to allow one to determine structures of solution sets of systems of algebraic difference-differential equations in many cases of interest. The proposed activity will result in the improvement of the existing algorithmic methods for PADDEs and more general systems of partial differential equations with group action, development of the constructive theory of difference and difference-differential ideals and, as a consequence, creation of new computational techniques for analysis of partial difference and difference-differential equations and their solution sets.The research will develop algorithms and computational techniques that will be of use to analysts, physicists, engineers, and scientists in many other fields where the theoretical description of processes involves algebraic differential, difference, or difference-differential equations. The resulting algorithms will be the basis of code appearing in symbolic computation computer packages used in education and research in mathematical physics, automatic control, mechanics, biology, and in many other areas as well. The educational component of the project also includes an interdisciplinary program that will involve mathematics, computer science, physics, and engineering majors in training and research with the active use of computer algebra methods.

本研究旨在开发用于偏代数差分微分方程组（PADDE）的计算分析及其解集描述的构造方法和算法。此类系统出现在数学及其应用中的各种问题中，包括数学物理、自动控制、动力系统、力学、分子化学和细胞生物学。本项目的主要研究目标是：（1）理论基础的发展以及差分-微分消除算法，特别是用于将 PADDE 系统的解集分解为“简单”集的并集； （2）将差分-微分代数的构造方法扩展到具有群作用的偏微分方程组（PDE）的计算分析（这对于应用程序特别感兴趣，因为控制物理场的 PDE 基本系统的解必须对于某些群体行为是不变的）； (3) 详细阐述用于计算表达爱因斯坦 PADDE 系统强度的维数多项式的方法和算法。特别是，此类算法将允许人们为物理过程的数学模型选择最佳（在爱因斯坦意义上）偏微分方程系统。要使用的主要方法和途径包括特征集技术，该技术将扩展到差分微分多项式环、差分微分模的广义 Groebner 基方法、单变量和多元维多项式技术以及 PADDE 的分解方法。尽管算法方法在 60 多年的历史中微分和差分代数，由 J. Ritt、E. Kolchin、R. Cohn 发起，最近由 M. Bronstein、X. Gau、P. Hendrics 和 M. Singer 等人扩展，目前还没有足够高效的计算方法允许人们在许多感兴趣的情况下确定代数差分微分方程组的解集的结构。所提议的活动将改进 PADDE 的现有算法方法和具有群作用的更一般的偏微分方程组，发展差分和差分微分理想的构造性理论，从而创建新的计算技术用于分析偏差分和差分微分方程及其解集。该研究将开发算法和计算技术，这些算法和计算技术将可用于许多其他领域的分析师、物理学家、工程师和科学家，其中过程的理论描述涉及代数微分,差分，或差分微分方程。由此产生的算法将成为出现在数学物理、自动控制、力学、生物学以及许多其他领域的教育和研究的符号计算计算机包中的代码的基础。 该项目的教育部分还包括一个跨学科项目，涉及数学、计算机科学、物理和工程专业，积极使用计算机代数方法进行培训和研究。