SBIR Phase I: Bifurcation analysis of nonlinear PDEs - a powerful Software Tool for Computational Design - Educational Applications

这一小型企业创新研究(SBIR)第一阶段研究项目将开发先进的软件工具，用于高效和准确地分析适用于教育应用计算设计等领域的非线性椭圆型偏微分方程(PDE)的数值分叉。非线性椭圆型偏微分方程组是许多科学和工程问题的基础，如生物中的化学反应、作用电势的模式形成和传播、生物物理学中的血液凝固级联等，在这些问题中，理解解对问题参数的定性依赖性是至关重要的。数值分支分析的主要方法是基于定义良好的算子方程的解的延拓。这样的计算结果使学生/研究人员对解的行为、稳定性、多重性和分叉有了更深入的了解，并经常提供基本数学理论的直接链接。交互界面和自动PDE离散化让学生/教师集中精力解决问题，提高了学习过程的效率。该活动的智能优点在于将基于径向函数(RBF)的偏微分方程组的高效、准确的离散方法集成到现有的数值分叉分析软件中。这一项目的结果将表明拟议概念的可行性，该概念可应用于各种问题。作为一个例子，这个软件工具将在一个重要的应用上得到验证--血凝级联的分析。目前市场上还没有用于教育应用的数值分叉分析工具。尽管无网格数值方法的最新进展为将这些方法整合到数值分支软件中创造了基础，但高精度的无网格方法将使研究人员能够优雅地识别用于非线性过程的数学描述的偏微分方程组的空间解。这些方法还可以帮助识别原本缺失的新制度，深入了解非线性系统动力学的演变，并提供更好的科学和技术理解。此外，如果该工具能够集成到广泛使用的数学工具--MATLAB平台环境中，这将使这些软件工具变得用户友好，可移植到所有操作系统，并允许对数据文件、图形输出等进行标准处理，并将使这些工具对大学各个理工系的教育具有吸引力。该项目还将通过启用软件分发的开放源码模式产生广泛的影响。

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 ofnonlinear 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.