AF: Small: Computational Methods for Difference-Differential Equations

本研究旨在发展偏代数差分微分方程组的计算分析及其解集描述的构造性方法和算法。这类系统出现在数学及其应用的各种问题中，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。本项目的主要研究目标是：(1)发展差异-微分消除的理论基础和算法，特别是将PADDEs系统的解集分解成“简单”集合的并；(2)将差分-微分代数的构造方法推广到具有群作用的偏微分方程组(PDE)的计算分析(这对于应用特别有意义，因为控制物理场的基本偏微分方程组的解必须关于某些群作用是不变的)；(3)详细阐述了表示爱因斯坦强度的维多项式的计算方法和算法。这些算法尤其允许人们为物理过程的数学模型选择最优的(在爱因斯坦意义上)PDE系统。要使用的主要方法和途径包括特征集技术，它将被扩展到差-微分多项式环，用于差-微分模块的广义Groebner基方法，一元和多变量维多项式的技术，以及用于PADE的分解方法。尽管微分和差分代数的算法方法的历史已经超过60年，由J.Ritt，E.Kolin，R.Cohn发起，最近由M.Bronstein，X.Gao，P.Hendrics和M.Singer扩展，在许多其他方法中，没有足够有效的计算方法来在许多感兴趣的情况下确定代数差微分方程组的解集的结构。这项拟议的活动将改进现有的偏微分方程组和更一般的具有群作用的偏微分方程组的算法方法，发展出关于差分和差分-微分理想的建构性理论，并由此产生用于分析偏差分和差分-微分方程组及其解集的新的计算技术。这项研究将开发算法和计算技术，将对许多其他领域的分析师、物理学家、工程师和科学家有用，这些领域的过程的理论描述涉及代数微分、差分或差分-微分方程组。由此产生的算法将成为出现在符号计算计算机包中的代码的基础，这些计算机包用于数学物理、自动控制、力学、生物学以及许多其他领域的教育和研究。该项目的教育部分还包括一个跨学科项目，该项目将让数学、计算机科学、物理和工程专业的学生积极使用计算机代数方法进行培训和研究。

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.