Least-Squares Solution of Linear Differential Equations
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 approac...
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| Format: | Article |
| Language: | English |
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MDPI AG
2017-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/5/4/48 |
| _version_ | 1818446433103118336 |
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| author | Daniele Mortari |
| author_facet | Daniele Mortari |
| author_sort | Daniele Mortari |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-14T19:47:38Z |
| format | Article |
| id | doaj.art-b716cdb9a9f840f1a321e888a8e7b30a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-14T19:47:38Z |
| publishDate | 2017-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-b716cdb9a9f840f1a321e888a8e7b30a2022-12-21T22:49:32ZengMDPI AGMathematics2227-73902017-10-01544810.3390/math5040048math5040048Least-Squares Solution of Linear Differential EquationsDaniele Mortari0Aerospace Engineering, Texas A&M University, College Station, TX 77843, USAThis 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.https://www.mdpi.com/2227-7390/5/4/48linear least-squaresinterpolationembedded linear constraints |
| spellingShingle | Daniele Mortari Least-Squares Solution of Linear Differential Equations Mathematics linear least-squares interpolation embedded linear constraints |
| title | Least-Squares Solution of Linear Differential Equations |
| title_full | Least-Squares Solution of Linear Differential Equations |
| title_fullStr | Least-Squares Solution of Linear Differential Equations |
| title_full_unstemmed | Least-Squares Solution of Linear Differential Equations |
| title_short | Least-Squares Solution of Linear Differential Equations |
| title_sort | least squares solution of linear differential equations |
| topic | linear least-squares interpolation embedded linear constraints |
| url | https://www.mdpi.com/2227-7390/5/4/48 |
| work_keys_str_mv | AT danielemortari leastsquaressolutionoflineardifferentialequations |