IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i9p1360-d1386086.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/9/1360/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/9/1360/
    Download Restriction: no
    ---><---

    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. 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.
    3. Daniele Mortari, 2017. "The Theory of Connections: Connecting Points," Mathematics, MDPI, vol. 5(4), pages 1-15, November.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Daniele Mortari & Carl Leake, 2019. "The Multivariate Theory of Connections," Mathematics, MDPI, vol. 7(3), pages 1-22, March.
    2. 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.
    3. Daniele Mortari, 2022. "Theory of Functional Connections Subject to Shear-Type and Mixed Derivatives," Mathematics, MDPI, vol. 10(24), pages 1-16, December.
    4. Yassopoulos, Christopher & Reddy, J.N. & Mortari, Daniele, 2023. "Analysis of nonlinear Timoshenko–Ehrenfest beam problems with von Kármán nonlinearity using the Theory of Functional Connections," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 709-744.
    5. Daniele Mortari & Roberto Garrappa & Luigi Nicolò, 2023. "Theory of Functional Connections Extended to Fractional Operators," Mathematics, MDPI, vol. 11(7), pages 1-18, April.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:9:p:1360-:d:1386086. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.