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.

中文翻译：

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分数扩展问题的近似分析

这项工作的目的是研究抽象贝塞尔型问题的近似，这是与拉普拉斯算子的分数幂相关的扩展问题的推广。受此类方法在分析抽象柯西问题的时间步长方法方面取得成功的启发，我们在此采用了类似的框架。所提出的方法不同于许多标准技术，因为我们近似于抽象问题的真实解决方案，而不是解决相关的离散问题。数值方法在适当的 Banach 空间中被证明是一致的、稳定的和收敛的。这些结果建立在半群理论的充分理解的结果之上。提供了数值实验来证明理论结果。