Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding

Hunter Johnston, Carl Leake, Yalchin Efendiev and Daniele Mortari
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Hunter Johnston: Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Carl Leake: Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Yalchin Efendiev: Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
Daniele Mortari: Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA

Mathematics, 2019, vol. 7, issue 6, 1-19

Abstract: 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.

Keywords: linear constraint optimization; calculus of variation; over-constrained differential equations; inequality constraints; triangular domains; Theory of Connections (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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