Theory of Functional Connections Applied to Linear Discontinuous Differential Equations

This article introduces two numerical methods based on the Theory of Functional Connections (TFC) for solving linear ordinary differential equations that involve step discontinuities in the forcing term. The novelty of the first proposed approach lies in the direct incorporation of discontinuities into the free function of the TFC framework, while the second proposed method resolves discontinuities through piecewise constrained expressions comprising particular weighted support functions systematically chosen to enforce continuity conditions. The accuracy of the proposed methods is validated for both a second-order initial value and boundary value problem. As a final demonstration, the methods are applied to a third-order differential equation with non-constant coefficients and multiple discontinuities, for which an analytical solution is known. The methods achieve error levels approaching machine precision, even in the case of equations involving functions whose Laplace transforms are not available.