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Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-Rectangular 2-Dimensional Domains.

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Summary

This study introduces bijective mappings to extend the Theory of Functional Connections (TFC) for non-rectangular domains. It presents novel mapping techniques and an inverse mapping approximation for broader TFC applications.

Keywords:
Theory of Functional Connectionsdomain mappingsfunctional interpolationleast-squares

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Area of Science:

  • Mathematics
  • Applied Mathematics
  • Computational Science

Background:

  • The Theory of Functional Connections (TFC) is a powerful tool for solving differential equations.
  • Current TFC methods are primarily limited to rectangular domains.
  • Extending TFC to non-rectangular domains is crucial for broader applicability.

Purpose of the Study:

  • To extend the Theory of Functional Connections (TFC) to handle non-rectangular two-dimensional domains.
  • To introduce and analyze novel bijective mapping techniques for this extension.
  • To develop methods for approximating inverse mappings when closed-form solutions are unavailable.

Main Methods:

  • Proposing three bijective mapping techniques: complex mapping, projection mapping, and polynomial mapping.
  • Developing a least-squares approximated inverse mapping for non-invertible mappings.
  • Demonstrating the replacement of piecewise boundary constraints with a single function.

Main Results:

  • Successfully extended TFC to non-rectangular domains using bijective mappings.
  • Introduced three distinct mapping strategies with analysis of their pros and cons.
  • Developed a functional approximation for inverse mappings, enhancing TFC's utility.
  • Showcased the simplification of boundary condition representation.

Conclusions:

  • Bijective mappings offer a viable approach to generalize TFC for complex geometries.
  • The proposed methods enhance the flexibility and scope of TFC applications.
  • This work lays the foundation for applying TFC to a wider range of scientific and engineering problems.