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

• Published in: Mathematical and computational applications Vol. 26; no. 3; p. 65
• Main Authors: De Florio, Mario, Schiassi, Enrico, D’Ambrosio, Andrea, Mortari, Daniele, Furfaro, Roberto
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2021
• Subjects: Basis functions; Chebyshev approximation; Constraints; Differential equations; Extreme Learning Machine; Functionals; Integrals; integro-differential equations; Mathematical models; Neural networks; numerical methods; Ordinary Differential Equations; Physics; Polynomials; Theory of Functional Connections
• ISSN: 2297-8747; 1300-686X; 2297-8747
• DOI: 10.3390/mca26030065

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.