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

Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
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Mathematics 2020, 8(3), 397; https://doi.org/10.3390/math8030397
Received: 10 February 2020 / Revised: 5 March 2020 / Accepted: 7 March 2020 / Published: 11 March 2020
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. View Full-Text
Keywords: differential equations; constraint embedding; theory of functional connections; boundary- value problems differential equations; constraint embedding; theory of functional connections; boundary- value problems
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Johnston, H.; Leake, C.; Mortari, D. Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections. Mathematics 2020, 8, 397.

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