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）和描述他们的解决方案集。这类系统出现在数学物理、自动控制、动力系统、力学、分子化学和细胞生物学等数学及其应用中，本项目的主要研究目标是：（1）发展差分-微分消去的理论基础和算法，特别是将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性方法推广到具有群作用的偏微分方程组的计算分析（这对应用特别有意义，因为支配物理场的偏微分方程的基本系统的解对于某些群作用必须是不变的）;（3）详细阐述了表示PADDE系统的爱因斯坦强度的维数多项式的计算方法和算法。这种算法，特别是，将允许一个选择最佳的（在A。Einstein）系统的偏微分方程的数学模型的物理过程。使用的主要方法和途径包括特征集技术，它将被扩展到差分-微分多项式环，差分-微分模的广义Groebner基方法，一元和多元维数多项式技术，尽管微分和差分代数的算法方法已有六十多年的历史，但由J. Ritt，E.科尔钦河Cohn和最近由M.布朗斯坦，X. Gao，P. Hendrics，and M.辛格，除其他外，有没有计算方法足够有效，让一个确定结构的解集系统的代数差分微分方程在许多情况下的利益。拟议的活动将导致改进现有的算法方法PADDE和更一般的系统偏微分方程与群作用，发展建设性理论的差异和差异微分理想，因此，创造新的计算技术，用于分析偏差和差-微分方程及其解集。这项研究将开发算法和计算技术，将用于分析师，物理学家，工程师，以及许多其他领域的科学家，其中过程的理论描述涉及代数微分，差分或差分微分方程。由此产生的算法将是出现在符号计算计算机软件包中的代码的基础，这些软件包用于数学物理、自动控制、力学、生物学和许多其他领域的教育和研究。 该项目的教育部分还包括跨学科的 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，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.