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A Theory of Functional Connections-Based hp -Adaptive Mesh Refinement Algorithm for Solving Hypersensitive Two-Point Boundary-Value Problems

Author

Listed:
  • Kristofer Drozd

    (System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA)

  • Roberto Furfaro

    (System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA)

  • Andrea D’Ambrosio

    (System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA)

Abstract

This manuscript introduces the first hp -adaptive mesh refinement algorithm for the Theory of Functional Connections (TFC) to solve hypersensitive two-point boundary-value problems (TPBVPs). The TFC is a mathematical framework that analytically satisfies linear constraints using an approximation method called a constrained expression. The constrained expression utilized in this work is composed of two parts. The first part consists of Chebyshev orthogonal polynomials, which conform to the solution of differentiation variables. The second part is a summation of products between switching and projection functionals, which satisfy the boundary constraints. The mesh refinement algorithm relies on the truncation error of the constrained expressions to determine the ideal number of basis functions within a segment’s polynomials. Whether to increase the number of basis functions in a segment or divide it is determined by the decay rate of the truncation error. The results show that the proposed algorithm is capable of solving hypersensitive TPBVPs more accurately than MATLAB R2021b’s bvp4c routine and is much better than the standard TFC method that uses global constrained expressions. The proposed algorithm’s main flaw is its long runtime due to the numerical approximation of the Jacobians.

Suggested Citation

  • Kristofer Drozd & Roberto Furfaro & Andrea D’Ambrosio, 2024. "A Theory of Functional Connections-Based hp -Adaptive Mesh Refinement Algorithm for Solving Hypersensitive Two-Point Boundary-Value Problems," Mathematics, MDPI, vol. 12(9), pages 1-35, April.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1360-:d:1386086
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    References listed on IDEAS

    as
    1. Kristofer Drozd & Roberto Furfaro & Enrico Schiassi & Andrea D’Ambrosio, 2023. "Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation," Mathematics, MDPI, vol. 11(17), pages 1-24, August.
    2. Daniele Mortari, 2017. "The Theory of Connections: Connecting Points," Mathematics, MDPI, vol. 5(4), pages 1-15, November.
    3. Binfeng Pan & Yang Wang & Shaohua Tian, 2018. "A High-Precision Single Shooting Method for Solving Hypersensitive Optimal Control Problems," Mathematical Problems in Engineering, Hindawi, vol. 2018, pages 1-11, April.
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