Least-Squares Method in the Theory of Matrix Differential-Algebraic Boundary-Value Problems

2018 ◽  
Vol 70 (2) ◽  
pp. 319-333
Author(s):  
S. M. Chuiko ◽  
O. V. Nesmelova ◽  
M. V. Dzyuba
Keyword(s):  
Least Squares ◽  
Boundary Value Problems ◽  
Least Squares Method ◽  
Boundary Value ◽  
Matrix Differential

scholarly journals Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 397 ◽  
Author(s):  
Hunter Johnston ◽  
Carl Leake ◽  
Daniele Mortari
Keyword(s):  
Differential Equation ◽  
Least Squares ◽  
Boundary Value Problems ◽  
Linear Combination ◽  
Boundary Value ◽  
Orthogonal Basis ◽  
Basis Functions ◽  
Eighth Order ◽  
Functional Connections ◽  
Free Function

This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a constrained expression) containing a function that the user is free to choose. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach.

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