The Multivariate Theory of Connections

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.