Analysis of an approximation to a fractional extension problem
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The purpose of this article 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.
KeywordsFractional diffusion Nonlocal operators Singular equations Degenerate equations Bessel equations Semigroup methods
Mathematics Subject Classification65J10 65M12 65R20 65M99
This work was supported by the NSF Grant Number 1903450. The author would like to thank Akif Ibraguimov, of Texas Tech University, for introducing this problem to them. The author is also thankful for Akif’s time and insightful suggestions that greatly improved the quality of the work herein. Finally, the author is thankful to the reviewers who provided numerous useful suggestions that greatly improved the presentation of the numerical experiments in Sect. 5.
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