The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations
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Publication:6344684
DOI10.3390/MATH8081303arXiv2007.04170WikidataQ113695235 ScholiaQ113695235MaRDI QIDQ6344684
Hunter Johnston, Daniele Mortari, Carl Leake
Publication date: 8 July 2020
Abstract: This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals--called constrained expressions--and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.
Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs (65M70) Generalized solutions to partial differential equations (35D99)
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