The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations

Carl Leake, Hunter Johnston and Daniele Mortari
Additional contact information
Carl Leake: Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Hunter Johnston: Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Daniele Mortari: Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA

Mathematics, 2020, vol. 8, issue 8, 1-30

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.

Keywords: Theory of Functional Connections; partial differential equations; least-squares; functional interpolation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/8/1303/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/8/1303/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2020:i:8:p:1303-:d:395329

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().