Stability in recovering the order of a time-fractional evolution equation
Abstract
We consider the recovery of the fractional order in a time-fractional evolution equation with a source of the form p(t)f, where the temporal factor p is also unknown but belongs to a bounded admissible class and has a prescribed nonzero value at t=0. From one scalar observation we prove a uniform small-time expansion and obtain a Hölder estimate for the fractional order. More precisely, if a fixed derivative order γ lies below every admissible fractional order, then the stability exponent is α+/(2α+−γ); for the undifferentiated observation this gives the exponent 1/2. We also show that differentiating the data to an order larger than all admissible fractional orders is supercritical: the corresponding derivative is unbounded near t=0 whenever the two fractional orders are different. A uniform estimate for the two-parameter Mittag--Leffler function, including the diagonal case, is proved in the appendix.
Related
- Recovering the Fractional Order and an Initial Value Simultaneously in a Time-Fractional Diffusion-Wave Equation
- Logarithmic stability for recovering initial data of fractional heat equations from thin-set observations
- Recovering the reaction coefficient in a time-fractional diffusion equation with variable coefficients in any spatial dimension
- Increasing stability for the inverse potential problem of the fractional Helmholtz equation
- Finite-Time Stability of Fractional-Order Damped Integro-Differential Systems with Impulsive Effects and Multiple Time Delays
BibTeX
@misc{ashurov2026stability,
title = {{Stability in recovering the order of a time-fractional evolution equation}},
author = {Ravshan Ashurov and Masahiro Yamamoto},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30799},
eprint = {2609.30799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30799},
}
RIS
TY - UNPB
TI - Stability in recovering the order of a time-fractional evolution equation
AU - Ravshan Ashurov
AU - Masahiro Yamamoto
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30799
UR - https://arxiv.org/abs/2609.30799
AB - We consider the recovery of the fractional order in a time-fractional evolution equation with a source of the form p(t)f, where the temporal factor p is also unknown but belongs to a bounded admissible class and has a prescribed nonzero value at t=0. From one scalar observation we prove a uniform small-time expansion and obtain a Hölder estimate for the fractional order. More precisely, if a fixed derivative order γ lies below every admissible fractional order, then the stability exponent is α_+/(2α_+−γ); for the undifferentiated observation this gives the exponent 1/2. We also show that differentiating the data to an order larger than all admissible fractional orders is supercritical: the corresponding derivative is unbounded near t=0 whenever the two fractional orders are different. A uniform estimate for the two-parameter Mittag--Leffler function, including the diagonal case, is proved in the appendix.
ER -