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.

这项研究旨在开发建设性方法和算法，以用于对部分代数差异方程（PADDES）系统系统的计算分析及其解决方案集的描述。这种系统在数学及其应用中出现了各种各样的问题，包括数学物理，自动控制，动力学系统，力学，分子化学和细胞生物学。该项目的关键研究目标是：（1）开发理论基础和算法，用于差异化的差异，尤其是分化解决方案的差异化解决方案集合的系统。 （2）将差异分别代数的建设性方法扩展到具有组动作的部分微分方程（PDE）系统的计算分析（这是应用程序的特别兴趣，因为PDES的基本系统的解决方案必须与某些小组行动相关的PDES系统的基本系统是不变的）; （3）详细阐述了计算尺寸多项式的方法和算法，这些多项式表达了爱因斯坦的强度。 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由差异和差异代数的算法方法的算法和差异方法的历史多六十年，由J. Ritt，E。Kolchin，R。Cohn和最近扩展，由M. Bronstein，X. Gao，X.Gao，P。在许多感兴趣的情况下，代数差异方程。所提出的活动将导致改善用于PADDE的算法方法和偏微分方程的更通用系统，并通过群体行动，发展差异和差异定义理想的建设性理论的发展，并因此而创建新的计算技术，用于分析部分差异和差异方程式和其解决方案的分析技术和计算的方法。在许多其他领域的物理学家，工程师和科学家，这些过程的理论描述涉及代数差异，差异或差异方程。由此产生的算法将是在数学物理学，自动控制，力学，生物学以及许多其他领域的教育和研究中使用的符号计算计算机软件包中出现的代码的基础。 该项目的教育组成部分还包括一个跨学科计划，该计划将涉及数学，计算机科学，物理和工程专业的培训和研究，并通过积极使用计算机代数方法。