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

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Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
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Department of Mathematics, Texas A&M University, College Station, TX 77843, USA
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Author to whom correspondence should be addressed.
Mathematics 2019, 7(6), 537; https://doi.org/10.3390/math7060537
Received: 14 May 2019 / Revised: 6 June 2019 / Accepted: 10 June 2019 / Published: 12 June 2019
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. View Full-Text
Keywords: linear constraint optimization; calculus of variation; over-constrained differential equations; inequality constraints; triangular domains; Theory of Connections linear constraint optimization; calculus of variation; over-constrained differential equations; inequality constraints; triangular domains; Theory of Connections
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Johnston, H.; Leake, C.; Efendiev, Y.; Mortari, D. Selected Applications of the Theory of Connections: A Technique for Analytical Constraint Embedding. Mathematics 2019, 7, 537.

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