This Small Business Innovation Research (SBIR) Phase I research project will develop advanced software tools for efficient and accurate numerical bifurcation analysis of nonlinear elliptic partial differential equations (PDEs) applicable to such fields as computational design for educational applications. Nonlinear elliptic PDEs are the basis for many scientific and engineering problems, such as chemical reactions, pattern formation and propagation of action potentials in biology, blood coagulation cascades in biophysics, etc. In these problems it is crucial to understand the qualitative dependence of the solution on the problem parameters. The principal approach of numerical bifurcation analysis is based on continuation of solutions to well-defined operator equations. Such computational results give to student/researcher a deeper understanding of the solution behavior, stability, multiplicity, and bifurcations, and often provide direct links to underlying mathematical theories. Interactive interface and automatic PDE discretization let the student/teacher concentrate on the problem solving, increasing the learning process efficiency. The intellectual merit of the proposed activity is in the integration of efficient and accurate discretization methods of PDEs by radial-based functions (RBFs) into existing numerical bifurcation analysis software. The outcome of this project will show the feasibility of the proposed concept that can be applied to a variety of problems. As an example, this software tool will be validated on an important application - the analysis of a blood coagulation cascade.
The numerical bifurcation analysis tools for educational applications are currently not available on the market. Although recent progress made in mesh less numerical methods creates a basis for incorporating these into numerical bifurcation software, high accuracy mesh less methods would allow researchers to elegantly identify the spatial solutions of the PDE systems that are used for mathematical description of nonlinear processes. These methods could also help to identify new regimes otherwise missing, to bring insights into evolution of nonlinear systems dynamics and provide enhanced scientific and technological understanding. Additionally, if this tool could be integrated into MATLAB platform environment - a widely used mathematics tool, which would enable these software tools to be user-friendly, portable to all operating systems and allow a standard handling of data files, graphical output, etc, and would allow these tools to be attractive for education at various science and engineering departments of universities. This project will also have a broad impact by enabling open source model for software distribution.
