# Analysis of an approximation to a fractional extension problem

## TL;DR

This paper investigates an approximation method for a fractional extension problem related to Bessel-type operators, demonstrating its consistency, stability, and convergence through theoretical analysis and numerical experiments.

## Contribution

It introduces a novel approximation approach for fractional extension problems, extending the framework of time-stepping methods to abstract Bessel-type problems.

## Key findings

- The method is consistent, stable, and convergent.
- Numerical experiments confirm theoretical predictions.
- The approach generalizes existing techniques for fractional operators.

## Abstract

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.