Analysis and Implementation of Nonlocal Governing Operators with Local Boundary Conditions on a General Interval

Abstract

We study nonlocal operators that enforce local boundary conditions on a general interval $(a,b)$. We prove the bijectivity of the operators in suitable spaces which leads to the well-posedness of the problems in the infinite dimensional setting. The operators are discretized using collocation methods. Their optimal convergence properties are established rigorously utilizing the theory of Fredholm operators of the second kind and the Hilbert-Schmidt property. The theoretical results are verified with numerical experiments.