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

Mathematical and Computational Applications

Daniele Mortari¹  Roberto Furfaro²  Enrico Schiassi²  Mario De Florio²  Andrea D’Ambrosio³ 

[1] Department of Aerospace Engineering, College of Engineering, Texas A&M University, 401 Joe Routt Blvd., College Station, TX 77843, USA;Department of Systems and Industrial Engineering, The University of Arizona, 1127 E. James E. Rogers Way, Tucson, AZ 85721, USA;School of Aerospace Engineering, Università degli Studi di Roma “La Sapienza”, Via Salaria 851, 00138 Rome, Italy;

关键词: Theory of Functional Connections;    Ordinary Differential Equations;    integro-differential equations;    Extreme Learning Machine;    numerical methods;   

DOI  :  10.3390/mca26030065

来源: DOAJ

【 摘 要 】

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, $g\left(x\right)$, representing the unknown function to optimize. Two numerical approaches are shown to numerically estimate $g\left(x\right)$. The first models $g\left(x\right)$ as a linear combination of a set of basis functions, such as Chebyshev or Legendre orthogonal polynomials, while the second models $g\left(x\right)$ as a neural network. Meaningful problems are provided. In all numerical problems, the proposed method produces very fast and accurate solutions.

【 授权许可】

Unknown