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.
The main project outcomes and findings in the reported period are as follows. 1. Development of the theory of characteristic sets for partial difference-differential polynomials in the case of several sets of operators and arbitrary integer powers of translations. The developed technique allowed the PI to obtain new methods and algorithms for solving several key problems of difference-differential algebra in this setting, such as whether an algebraic difference-differential equation is a consequence of a given system of such equations or whether a system of algebraic difference-differential equations is consistent. Another application of the new method of characteristic sets, which involves several rankings of indeterminates, is the development of new algorithms for the computation of multivariate dimension polynomials that express the A. Einstein’s strength of systems of algebraic difference-differential equations with several sets of basic difference operators and their inverses. 2. Development of constructive methods for building mixed difference and difference-differential polynomial ideals. The PI has found a scheme for constructing mixed difference and difference-differential ideals and their radicals. In addition to a new constructive approach to the description of solution sets of algebraic difference and difference-differential equations, these results allowed PI to solve an open problem on the maximality condition for mixed difference ideals. 3. Generalization of the relative Gröbner basis technique to modules over rings of inversive difference-differential operators and Weyl algebras. Elaboration of algorithms of computation of multivariate difference-differential dimension polynomials based on the relative Gröbner basis method. 4. Computation of invariants of multivariate dimension polynomials and application of the properties of such invariants to the isomorphism problem for difference-differential modules and systems of difference-differential equations. In particular, the PI has obtained new necessary conditions for the equivalence of two systems of algebraic difference-differential equations in the sense of A. Einstein. 5. Computation of the strength of fundamental PDEs of mathematical physics (e. g., Maxwell equations and equations for electromagnetic field given by potential), as well as the strength of systems of difference equations obtained from these PDEs via different difference schemes. Comparison of different difference schemes by their Einstein’s strength. Students and human resources development: The PI's graduate student Christian Dönch successfully defended his PhD dissertation Standard Bases in Finitely Generated Difference-Skew-Differential Modules and Their Application to Dimension Polynomials on July 31, 2012. In the frame of the project activity, the PI developed syllabi, complete sets of homework assignments, and sample exam problems for his new CUA courses, an undergraduate course Methods of Computer Algebra and a graduate course Computational Difference-Differential Algebra designed for the CUA mathematics, computer science and physics majors. Broader Impact: The research activity in the reported period provides new tools for analysis of systems of ordinary and partial difference and difference-differential equations. The new technique will be important for many applications. It will be used In physics, in the study of fundamental systems of partial difference and difference-differential equations that involve arbitrary integer powers of basic difference operators. In engineering, in the study and development of continuous and discrete systems of automatic control. In biology, in the analysis of continuous and discrete biological models, as well as in other areas where mathematical models of the processes are described by systems of difference-differential equations. The research is naturally integrated into the interdepartmental collaboration and joint educational programs for computer science, mathematics and physics majors, as well as for engineering majors in the field of automatic control and biology majors who work with continuous and discrete mathematical models of biology systems.
