Next Article in Journal
The Evolution of Mathematical Thinking in Chinese Mathematics Education
Previous Article in Journal
Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety
Article Menu
Issue 3 (March) cover image

Export Article

Open AccessFeature PaperArticle
Mathematics 2019, 7(3), 296;

The Multivariate Theory of Connections

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
Authors to whom correspondence should be addressed.
This paper is an extended version of our paper published in Mortari, D. “The Theory of Connections: Connecting Functions.” IAA-AAS-SciTech-072, Forum 2018, Peoples’ Friendship University of Russia, Moscow, Russia, 13–15 November 2018.
Received: 4 January 2019 / Revised: 25 February 2019 / Accepted: 18 March 2019 / Published: 22 March 2019
(This article belongs to the Special Issue Computational Mathematics, Algorithms, and Data Processing)
PDF [3000 KB, uploaded 22 March 2019]
  |     |  


This paper extends the univariate Theory of Connections, introduced in (Mortari, 2017), to the multivariate case on rectangular domains with detailed attention to the bivariate case. In particular, it generalizes the bivariate Coons surface, introduced by (Coons, 1984), by providing analytical expressions, called constrained expressions, representing all possible surfaces with assigned boundary constraints in terms of functions and arbitrary-order derivatives. In two dimensions, these expressions, which contain a freely chosen function, g ( x , y ) , satisfy all constraints no matter what the g ( x , y ) is. The boundary constraints considered in this article are Dirichlet, Neumann, and any combinations of them. Although the focus of this article is on two-dimensional spaces, the final section introduces the Multivariate Theory of Connections, validated by mathematical proof. This represents the multivariate extension of the Theory of Connections subject to arbitrary-order derivative constraints in rectangular domains. The main task of this paper is to provide an analytical procedure to obtain constrained expressions in any space that can be used to transform constrained problems into unconstrained problems. This theory is proposed mainly to better solve PDE and stochastic differential equations. View Full-Text
Keywords: interpolation; constraints; embedded constraints interpolation; constraints; embedded constraints

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

Share & Cite This Article

MDPI and ACS Style

Mortari, D.; Leake, C. The Multivariate Theory of Connections. Mathematics 2019, 7, 296.

Show more citation formats Show less citations formats

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

Related Articles

Article Metrics

Article Access Statistics



[Return to top]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top