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
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,
, and they satisfy the constraints, no matter what
is. The second step consists of expressing
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,
. 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.
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MDPI and ACS Style
Mortari, D. Least-Squares Solution of Linear Differential Equations. Mathematics 2017, 5, 48.
Mortari D. Least-Squares Solution of Linear Differential Equations. Mathematics. 2017; 5(4):48.
Mortari, Daniele. 2017. "Least-Squares Solution of Linear Differential Equations." Mathematics 5, no. 4: 48.
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