Abstract
Based on the popular Caputo fractional derivative of order β in (0, 1), we define the censored fractional derivative on the positive half-line ℝ+. This derivative proves to be the Feller generator of the censored (or resurrected) decreasing β-stable process in ℝ+. We provide a series representation for the inverse of this censored fractional derivative. We are then able to prove that this censored process hits the boundary in a finite time τ∞, whose expectation is proportional to that of the first passage time of the β-stable subordinator. We also show that the censored relaxation equation is solved by the Laplace transform of τ∞. This relaxation solution proves to be a completely monotone series, with algebraic decay one order faster than its Caputo counterpart, leading, surprisingly, to a new regime of fractional relaxation models. Lastly, we discuss how this work identifies a new sub-diffusion model.
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Du, Q., Toniazzi, L. & Xu, Z. Censored Stable Subordinators and Fractional Derivatives. Fract Calc Appl Anal 24, 1035–1068 (2021). https://doi.org/10.1515/fca-2021-0045
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DOI: https://doi.org/10.1515/fca-2021-0045