Least-squares Solutions of Eighth-order Boundary Value Problems using the Theory of Functional Connections
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Publication:6334690
DOI10.3390/MATH8030397arXiv2002.05113WikidataQ96025570 ScholiaQ96025570MaRDI QIDQ6334690
Daniele Mortari, Hunter Johnston, Carl Leake
Publication date: 10 February 2020
Abstract: 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 ) that contains 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.
Numerical interpolation (65D05) Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations (65L60) Numerical solution of boundary value problems involving ordinary differential equations (65L10) Boundary value problems for functional-differential equations (34K10)
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