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The Multivariate Theory of Connections.

Daniele Mortari1, Carl Leake1

  • 1Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA.

Mathematics (Basel, Switzerland)
|June 2, 2020
PubMed
Summary

This study introduces the multivariate Theory of Connections, generalizing bivariate surfaces with analytical expressions for all boundary constraints. This method transforms constrained problems into unconstrained ones, aiding in solving partial differential equations (PDEs) and stochastic differential equations.

Keywords:
constraintsembedded constraintsinterpolation

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Area of Science:

  • Multivariate Calculus
  • Differential Geometry
  • Numerical Analysis

Background:

  • The univariate Theory of Connections provides a framework for boundary value problems.
  • The bivariate Coons surface is a foundational concept for interpolating surfaces with boundary constraints.
  • Existing methods often struggle with arbitrary-order derivative constraints and diverse boundary conditions.

Purpose of the Study:

  • To extend the univariate Theory of Connections to the multivariate case on rectangular domains.
  • To generalize the bivariate Coons surface by developing analytical 'constrained expressions' that encompass all possible solutions for given boundary constraints.
  • To provide a universal analytical procedure for transforming constrained problems into unconstrained ones, applicable to various mathematical and scientific domains.

Main Methods:

  • Generalization of the univariate Theory of Connections to multivariate spaces.
  • Development of analytical 'constrained expressions' for bivariate surfaces satisfying Dirichlet, Neumann, and mixed boundary conditions.
  • Mathematical proof validating the Multivariate Theory of Connections for arbitrary-order derivative constraints in rectangular domains.

Main Results:

  • Analytical expressions ('constrained expressions') are derived, representing all surfaces with assigned boundary constraints, including arbitrary-order derivatives.
  • These expressions incorporate a freely chosen function, ensuring satisfaction of all boundary constraints regardless of the function's form.
  • The Multivariate Theory of Connections is formally established, extending the theory to higher dimensions and arbitrary derivative constraints.

Conclusions:

  • The developed theory and constrained expressions offer a unified approach to handling boundary constraints in multivariate settings.
  • This framework has significant implications for efficiently solving partial differential equations (PDEs) and stochastic differential equations.
  • The analytical procedure facilitates the transformation of complex constrained problems into simpler unconstrained ones, advancing computational mathematics.