Generalized artificial-time PDE formulations and computational techniques for multi-rate systems

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A novel theoretical/computational approach, termed the Generalized Multitime Partial Differential Equation (GeMPDE), is being investigated. The chief advantage of the GeMPDE formulation is that it fundamentally enables, speeds up or greatly simplifies theoretical and computational analysis of physical and engineering systems, which feature slow/fast rates of dynamical change separated by many orders of magnitude.Differentiating features and intellectual merits that make the GeMPDE considerably more attractive than existing approaches include: its ability (demonstrated in preliminary results) to perform linearized .AC analysis of oscillatory systems with full generality and rigor (not possible previously); its use of multivariate phase and local frequency variables to yield theoretically and computationally robust system formulations; and its clean separation of disparate rates of variation using separated artificial time variables.Core theoretical and computational tasks being explored include: stochastic GeMPDE analysis, investigation of specialized GeMPDE forms targeting specific application domains, the development of robust GeMPDE-based solvers that adapt automatically to nonlinear systems containing both oscillatory and non-oscillatory components, large signal GeMPDE-based envelope analysis, and applications to periodically forced systems, systems of coupled oscillators and non-periodic autonomous systems.Broader impacts include a strongly linking educational and research components; direct application/validation on electronic circuits (PLLs, SD modulators), nano technological devices (arrays of Josephson junctions and single-electron tunneling devices), and biological systems (networks of firing neurons); and the use of open source methodologies for developing and disseminating prototype code resulting from this research.