Next Article in Journal
Cokriging Prediction Using as Secondary Variable a Functional Random Field with Application in Environmental Pollution
Previous Article in Journal
Improving Stability Conditions for Equilibria of SIR Epidemic Model with Delay under Stochastic Perturbations
Article

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

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(8), 1303; https://doi.org/10.3390/math8081303
Received: 8 July 2020 / Revised: 27 July 2020 / Accepted: 4 August 2020 / Published: 6 August 2020
(This article belongs to the Section Difference and Differential Equations)
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. View Full-Text
Keywords: Theory of Functional Connections; partial differential equations; least-squares; functional interpolation Theory of Functional Connections; partial differential equations; least-squares; functional interpolation
Show Figures

Figure 1

MDPI and ACS Style

Leake, C.; Johnston, H.; Mortari, D. The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations. Mathematics 2020, 8, 1303. https://doi.org/10.3390/math8081303

AMA Style

Leake C, Johnston H, Mortari D. The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations. Mathematics. 2020; 8(8):1303. https://doi.org/10.3390/math8081303

Chicago/Turabian Style

Leake, Carl, Hunter Johnston, and Daniele Mortari. 2020. "The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations" Mathematics 8, no. 8: 1303. https://doi.org/10.3390/math8081303

Find Other Styles
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Article Access Map by Country/Region

1
Back to TopTop