Functional Interpolation with Prescribed Singularities: Extending the Theory of Functional Connections to Divergent Constraints

This paper extends the Theory of Functional Connections (TFC) to a new class of admissibility conditions, referred to as divergent constraints, which prescribe an unbounded asymptotic behavior of the solution at one or more specified locations. While the existing TFC literature has established a comprehensive treatment of pointwise, derivative, integral, and general linear-operator constraints, no systematic framework was available for constraints that force the solution to diverge at prescribed points. The main contribution of this work is a rigorous univariate formulation that simultaneously embeds an arbitrary number of regular linear constraints and divergent constraints into a single analytical functional, together with a proof of a decoupling property that allows the regular and divergent switching functions to be computed by two independent families of linear systems. A singularity-aware quadrature is also introduced to evaluate the integrals arising in the assembly of the divergent-switching-function systems to machine precision. The theoretical developments are supported by numerical experiments that verify the prescribed regular and divergent behavior for several representative constraint configurations.