Fixed Point Theory, Nonlinear Differential Equations and Computational Algorithms on Data Analytics

点击翻译按钮获取中文摘要

The proposed research overlaps with three closely related areas in applied mathematics and computing: fixed point theory, nonlinear differential equations and computational algorithms for data analytics. Each topic on its own is important and substantial. Connecting the three areas together (fixed-point theory, nonlinear differential equations, and neural networks) along a continuum from theory to real-world applications is innovative and the central focus of my research.*******As a very powerful and important tool in studying differential equations, fixed point theory has diverse applications in various fields such as Biology, Chemistry, Economics, Engineering, Game Theory, Physics, and Computer Sciences. The research lies in the development and application of new mathematical techniques in fixed point theory to study solutions for nonlinear differential equations arising from population dynamics, reaction-diffusion systems, image classifications, neural networks and social activities. In particular, systems of differential equations involving multiple parameters with non-local boundary or initial conditions will be considered. In applications, this class of differential systems represents gas diffusion, heat conduction, elastic beam and in general feedback controls by which the “sum” of the states of the process along its evolution equals the initial state in real processes from Physics, Chemistry or Biology. ******In addition to topological methods from functional analysis, nonlinear analysis and linear operator theory, constructional techniques such as iteration, monotone mappings, upper and lower solutions will also be applied. On the computational side, new algorithms will be developed, implemented and tested using real-world data sets. System performance will be evaluated and documented. Computational approaches including numerical methods and computer simulations will be applied. New tools and methods will also be developed.******The results will contribute across several fields in nonlinear analysis, dynamical systems, mathematical modeling and computational algorithms for data analytics. The last field, in particular, has received considerable attention of late. Our work will continue to contribute to the well-being of our Canadian research community, both in fundamental and applied terms.********