NDLI: Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

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Theory of Functional Connections Applied to Linear ODEs Subject to Integral Constraints and Linear Ordinary Integro-Differential Equations

Content Provider	MDPI
Author	D’Ambrosio, Andrea Florio, Mario De Schiassi, Enrico Mortari, Daniele Furfaro, Roberto
Copyright Year	2021
Description	This study shows how the Theory of Functional Connections (TFC) allows us to obtain fast and highly accurate solutions to linear ODEs involving integrals. Integrals can be constraints and/or terms of the differential equations (e.g., ordinary integro-differential equations). This study first summarizes TFC, a mathematical procedure to obtain constrained expressions. These are functionals representing all functions satisfying a set of linear constraints. These functionals contain a free function, $g(x)$ , representing the unknown function to optimize. Two numerical approaches are shown to numerically estimate $g(x)$ . The first models $g(x)$ as a linear combination of a set of basis functions, such as Chebyshev or Legendre orthogonal polynomials, while the second models $g(x)$ as a neural network. Meaningful problems are provided. In all numerical problems, the proposed method produces very fast and accurate solutions.
Starting Page	65
e-ISSN	22978747
DOI	10.3390/mca26030065
Journal	Mathematical and Computational Applications
Issue Number	3
Volume Number	26
Language	English
Publisher	MDPI
Publisher Date	2021-09-12
Access Restriction	Open
Subject Keyword	Mathematical and Computational Applications Mathematical Physics Theory of Functional Connections Ordinary Differential Equations Integro-differential Equations Extreme Learning Machine Numerical Methods
Content Type	Text
Resource Type	Article

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