AF: Small: Computational Methods for Difference-Differential Equations

本研究的目的是发展建设性的方法和算法的计算分析系统的偏代数差分-微分方程（PADDEs）和描述其解集。这样的系统出现在各种各样的数学问题及其应用中，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)发展差分-微分消元的理论基础和算法，特别是将pads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推广到具有群作用的偏微分方程（PDEs）系统的计算分析（这对于应用来说是特别有趣的，因为控制物理场的PDEs基本系统的解对于某些群作用必须是不变的）；(3)详细阐述了维多项式的计算方法和算法，这些维多项式表达了爱因斯坦的PADDEs系统的强度。特别是，这样的算法将允许人们为物理过程的数学模型选择最优（在爱因斯坦的意义上）偏微分方程系统。主要使用的方法和途径包括特征集技术（将扩展到差分-微分多项式环）、差分-微分模块的广义Groebner基方法、单维和多维多项式技术以及微分方程的分解方法。尽管微分和差分代数的算法方法有超过60年的历史，由J. Ritt， E. Kolchin， R. Cohn发起，最近由M. Bronstein， X. Gao， P. Hendrics和M. Singer等人扩展，但在许多感兴趣的情况下，没有足够有效的计算方法允许人们确定代数微分方程系统解集的结构。拟议的活动将导致改进现有的微分方程算法方法和具有群作用的更一般的偏微分方程系统，发展差分和差分-微分理想的建设性理论，并因此创造新的计算技术来分析偏差分和差分-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用于分析人员、物理学家、工程师和许多其他领域的科学家，这些领域的理论描述过程涉及代数微分、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、生物学和许多其他领域的教育和研究中使用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，将涉及数学，计算机科学，物理和工程专业的培训和研究，并积极使用计算机代数方法。

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.