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

TL;DR

This paper applies the Theory of Functional Connections to develop a least-squares method for solving boundary-value problems in hybrid systems, enabling efficient and accurate solutions for both linear and nonlinear differential equations.

Contribution

It introduces an analytical constrained expression for hybrid systems that enforces boundary and continuity conditions, reducing the solution space to admissible solutions and enabling least-squares solutions.

Findings

Achieves machine-error accuracy in numerical tests

Validates the approach on linear and nonlinear differential equations

Provides a general formulation for n-segment hybrid systems

Abstract

This paper looks to apply the mathematical framework of the Theory of Functional Connections to the solution of boundary-value problems arising from hybrid systems. The Theory of Functional Connections is a technique to derive constrained expressions which are analytical expressions with embedded constraints. These expressions are particularly suitable 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). A hybrid system is simply a sequence of different differential equations. 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 a iterative least-squares solution for nonlinear differential equations. Lastly, the general formulation for "n" segments is developed and validation is provided through numerical tests for three differential equation sequences: linear/linear, linear/nonlinear, and nonlinear/nonlinear. The accuracy level 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.