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New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations

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Mathematics 2017, 5(4), 48; https://doi.org/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

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

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