Symbolic computation for differential equation based systems

这个项目是在符号计算动力学领域。这是一个新的方法来建模和解决微分方程为基础的问题，在科学和工程。符号计算动力学的学术根源可以追溯到七十年代用MACSYMA语言编写的半自动微分方程求解代码的早期实现。从那时起，申请人和其他与他一起工作的人在这一领域进行了大量的研究工作，我们现在称之为符号计算动力学的方法在过去的25年里已经得到了我们的发展。该工作的很大一部分由EPSRC（Cartmell PI）资助。一家初创公司Symbolic Computational Dynamics Ltd（SCD Ltd）于2020年6月在格拉斯哥注册成立，以推动产品开发工作。SCD有限公司资助这个学生在£12680的水平，因为有一个平行的和重要的学术工作要做，这是需要支持正在进行的产品开发活动在公司.符号计算动力学的主题是基于这样一个事实，即所有现实世界的动力系统是数学非线性的，并且通过按比例缩小和线性化的建模简化只能有有限的用途，因为通常只有最低阶的现象才能以这种方式被捕获。有限元分析潜在地提供了一种通用的和强大的方法，但在一般性上受到FEA模型所基于的问题的特定数值表示的限制。这意味着，如果需要对问题域进行全局理解，则FEA在计算上变得昂贵。符号计算动力学完全基于微分方程，无论是常非线性微分方程系统，还是偏非线性微分方程系统和边界条件，并使用渐近分析构造这些方程的近似解析解。传统上，这种非线性渐近分析是手工完成的，并且对于非常小的微分方程系统，因为代数复杂性和所需的纯粹努力。现在，这一切都可以在计算机上完成，运行该团队在过去25年中开发的计算求解器代码，并根据需要提供不同的准确度。这套求解器的最新版本能够在几分钟内完成对二阶甚至更高阶扰动近似的整个渐近分析。这是通过在基于云的服务器应用程序上运行未优化的串行代码来实现的，因此将研究加速技术。除此之外，我们还将构建新一代用户界面（UI），以实现与用户的交互。

This project is in the area of symbolic computational dynamics. This is a new approach to the modelling and solution of differential equation-based problems in science and engineering. The academic roots of symbolic computational dynamics can be traced back to some early implementations of semi-automated differential equation solution code written in the MACSYMA language in the seventies. Since then a lot of research work has been done in this area by the applicant and others working with him, and the approach which we now call symbolic computational dynamics has been evolved by us over the last 25 years. A significant proportion of that work was funded by EPSRC (Cartmell PI) A start-up company, Symbolic Computational Dynamics Ltd (SCD Ltd) was incorporated in Glasgow in June 2020 to drive the product development work. SCD Ltd is funding this studentship at a level of £12680 because there is a parallel and vital strand of academic work to be done which is needed to support the product development activity being undertaken in the Company.The topic of Symbolic Computational Dynamics is based on the fact that all real-world dynamical systems are mathematically nonlinear, and that modelling simplifications through scaling down and linearisation can only ever be of limited use because usually only the lowest order phenomena are captured that way. Finite Element Analysis potentially offers a versatile and powerful way round this problem but is limited in generality by the specific numerical representation of the problem upon which the FEA model is based. This means that FEA becomes computationally expensive if a global understanding of the problem domain is required. Symbolic Computational Dynamics is based entirely on differential equations, either systems of ordinary nonlinear differential equations, or systems of partial nonlinear differential equations and boundary conditions, and constructing approximate analytical solutions to these equations using asymptotic analysis. Traditionally this sort of nonlinear asymptotic analysis is done by hand and for very small systems of differential equations, because of the algebraic complexity and the sheer effort needed. Now this can all be done on a computer, running the computational solver codes that have been developed by the team over the last 25 years, and with different levels of accuracy as required. The latest version of this suite of solvers is capable of doing an entire asymptotic analysis to second or even higher order perturbation approximation in a matter of a few minutes. This is achieved by running un-optimised serial code on a cloud-based server application, so speed-up techniques will be researched. In addition to this we will build a new generation of user interfaces (UIs) to enable interaction with the user.