Least-squares solutions of boundary-value problems in hybrid systems

Abstract

This paper applies the mathematical framework of the Theory of Functional Connections to solve the boundary-value problems arising from hybrid systems. A hybrid system is simply a sequence of different differential equations. The Theory of Functional Connections is a technique to derive constrained expressions which are functionals with embedded constraints. These functionals allow to transform a large class of constrained optimization problems into unconstrained problems. The initial and most useful application of this technique is in the solution of differential equations where the problem can be posed as an unconstrained optimization problem and solved with simple numerical techniques, i.e., least-squares.The approach developed in this work derives an analytical constrained expression for the entire range of a hybrid system, enforcing both the boundary conditions as well as the continuity conditions across the sequence of differential equations. This reduces the searched solution space of the hybrid system to only admissible solutions. The transformation allows for a least-squares solution of the sequence for linear differential equations and an iterative least-squares solution for nonlinear differential equations.This technique is validated by numerical tests for three differential equation sequences: linear–linear, linear–nonlinear, and nonlinear–nonlinear. The accuracy levels obtained are all at machine-error, which is consistent with the accuracy experienced in past studies on the application of the Theory of Functional Connections to solve single ordinary differential equations. In these tests, TFC produces solutions up to 3 orders of magnitude more accurate than the spectral method and up to 4 orders of magnitude more accurate than the shooting method. Additionally, in all cases except the linear–linear sequence, the TFC method is faster. Finally, the proposed framework is used to solve the one-dimensional convection–diffusion equation to further highlight its utility outside of hybrid systems.