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Mathematics 2017, 5(4), 48; doi:10.3390/math5040048

Least-Squares Solution of Linear Differential Equations

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
This paper is an extended version of our paper published in Mortari, D. Least-squares Solutions of Linear Differential Equations. In Proceedings of 27th AAS/AIAA Space Flight Mechanics Meeting Conference, San Antonio, TX, USA, 5–9 February 2017.
Received: 30 July 2017 / Revised: 11 September 2017 / Accepted: 29 September 2017 / Published: 8 October 2017
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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. View Full-Text
Keywords: linear least-squares; interpolation; embedded linear constraints linear least-squares; interpolation; embedded linear constraints

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Mortari, D. Least-Squares Solution of Linear Differential Equations. Mathematics 2017, 5, 48.

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