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Least-Squares Solution of Linear Differential Equations

Author

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  • Daniele Mortari

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

Abstract

This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression , that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [ − 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.

Suggested Citation

  • Daniele Mortari, 2017. "Least-Squares Solution of Linear Differential Equations," Mathematics, MDPI, vol. 5(4), pages 1-18, October.
  • Handle: RePEc:gam:jmathe:v:5:y:2017:i:4:p:48-:d:114308
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    Citations

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    Cited by:

    1. Enrico Schiassi & Mario De Florio & Andrea D’Ambrosio & Daniele Mortari & Roberto Furfaro, 2021. "Physics-Informed Neural Networks and Functional Interpolation for Data-Driven Parameters Discovery of Epidemiological Compartmental Models," Mathematics, MDPI, vol. 9(17), pages 1-17, August.
    2. 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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