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The Theory of Connections: Connecting Points

Author

Listed:
  • Daniele Mortari

    (Aerospace Engineering, Texas A & M University, College Station, TX 77843-3141, USA)

Abstract

This study introduces a procedure to obtain all interpolating functions, y = f ( x ) , subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through a single point in three distinct ways: linear, additive, and rational. Then, using the additive formalism, interpolating functions with linear constraints on one, two, and n points are introduced as well as those satisfying relative constraints. In particular, for expressions passing through n points, a generalization of the Waring’s interpolation form is introduced. An alternative approach to derive additive constraint interpolating expressions is introduced requiring the inversion of a matrix with dimensions equally the number of constraints. Finally, continuous and discontinuous interpolating periodic functions passing through a set of points with specified periods are provided. This theory has already been applied to obtain least-squares solutions of initial and boundary value problems applied to nonhomogeneous linear differential equations with nonconstant coefficients.

Suggested Citation

  • Daniele Mortari, 2017. "The Theory of Connections: Connecting Points," Mathematics, MDPI, vol. 5(4), pages 1-15, November.
    • Handle: RePEc:gam:jmathe:v:5:y:2017:i:4:p:57-:d:117161
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    Citations

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    Cited by:

    1. Daniele Mortari, 2022. "Theory of Functional Connections Subject to Shear-Type and Mixed Derivatives," Mathematics, MDPI, vol. 10(24), pages 1-16, December.
    2. Daniele Mortari & Carl Leake, 2019. "The Multivariate Theory of Connections," Mathematics, MDPI, vol. 7(3), pages 1-22, March.
    3. 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.
    4. 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.
    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.
    6. 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.

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