  The purpose of this work is to study an approximation to an abstract
Bessel-type problem, which is a generalization of the extension problem
associated with fractional powers of the Laplace operator. Motivated by the
success of such approaches in the analysis of time-stepping methods for
abstract Cauchy problems, we adopt a similar framework, herein. The proposed
method differs from many standard techniques, as we approximate the true
solution to the abstract problem, rather than solve an associated discrete
problem. The numerical method is shown to be consistent, stable, and convergent
in an appropriate Banach space. These results are built upon well understood
results from semigroup theory. Numerical experiments are provided to
demonstrate the theoretical results.
