Johnston Hunter, Leake Carl, Efendiev Yalchin, Mortari Daniele
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA.
Mathematics (Basel). 2019 Jun;7(6):537. doi: 10.3390/math7060537. Epub 2019 Jun 12.
In this paper, we consider several new applications of the recently introduced mathematical framework of the Theory of Connections (ToC). This framework transforms constrained problems into unconstrained problems by introducing constraint-free variables. Using this transformation, various ordinary differential equations (ODEs), partial differential equations (PDEs) and variational problems can be formulated where the constraints are always satisfied. The resulting equations can then be easily solved by introducing a global basis function set (e.g., Chebyshev, Legendre, etc.) and minimizing a residual at pre-defined collocation points. In this paper, we highlight the utility of ToC by introducing various problems that can be solved using this framework including: (1) analytical linear constraint optimization; (2) the brachistochrone problem; (3) over-constrained differential equations; (4) inequality constraints; and (5) triangular domains.
在本文中，我们考虑了最近引入的联络理论（ToC）数学框架的几个新应用。该框架通过引入无约束变量将约束问题转化为无约束问题。利用这种变换，可以构建各种常微分方程（ODE）、偏微分方程（PDE）和变分问题，其中约束条件总能得到满足。然后，通过引入一个全局基函数集（例如，切比雪夫、勒让德等）并在预定义的配置点处最小化残差，就可以轻松求解所得方程。在本文中，我们通过引入各种可以使用此框架解决的问题来突出ToC的实用性，这些问题包括：（1）解析线性约束优化；（2）最速降线问题；（3）超定微分方程；（4）不等式约束；以及（5）三角形区域。
