Abstract
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.
Highlights
This paper shows how to solve linear Ordinary Differential Equations (ODEs) and linear Integro-Differential Equations (IDEs) using a new mathematical framework to perform functional interpolation, called Theory of Functional Connections (TFC)
The free function can be expressed by any set of linearly independent functions, such as an expansion of orthogonal polynomials (e.g., Chebyshev, Legendre, etc.) or Neural Networks (NN), such as shallow NN with random features, or Deep NNs (DNNs)
When shallow NNs with random features are used, the method has been identified as Extreme-TFC (X-TFC) [9], and when DNNs are used the method is called Deep-TFC [10]
Summary
This paper shows how to solve linear Ordinary Differential Equations (ODEs) and linear Integro-Differential Equations (IDEs) using a new mathematical framework to perform functional interpolation, called Theory of Functional Connections (TFC). TFC derives functionals, called constrained expressions, containing a free function and representing all possible functions satisfying a set of linear constraints [1,2,3,4]. The most important feature of the constrained expressions is: they always satisfy all the constraints no matter what the free function is It was recently developed (2017), TFC has already found several applications, especially in solving differential equations [5,6,7,8]. The free function can be expressed by any set of linearly independent functions, such as an expansion of orthogonal polynomials (e.g., Chebyshev, Legendre, etc.) or Neural Networks (NN), such as shallow NN with random features, or Deep NNs (DNNs). Since the classic TFC uses Chebyshev or Legendre orthogonal polynomials as a basis set, the final Appendix A provides a definition, orthogonality, derivatives, and integral expressions and properties for both Chebyshev and Legendre orthogonal polynomials
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